Class 9 Maths — Important Questions with Answers

57 concept-first Class 9 Maths questions across 13 chapters — each with a clear model answer, the why behind it, and a memory trick to make it stick. NCERT-aligned and free. Build the base now, and your board exams, NEET and JEE feel a whole lot easier later.

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Number Systems (4)Polynomials (5)Linear Equations in Two Variables (3)Heron's Formula (4)Surface Areas and Volumes (18)Statistics (6)Triangles (5)Mensuration (3)Lines and Angles (3)Coordinate Geometry (1)Quadrilaterals (2)Circles (2)Probability (1)

Number Systems4 questions

MathematicsRationalising the denominatormedium

Rationalise the denominator of 1 / sqrt(2).

Reveal answer▾

What it is

Rationalising clears a surd (root) from the denominator by multiplying top and bottom by it.

Answer

Multiply the numerator and denominator by sqrt(2): (1/sqrt(2)) x (sqrt(2)/sqrt(2)) = sqrt(2) / 2. So 1/sqrt(2) = sqrt(2)/2, which is about 0.707. Rationalising removes the irrational number (surd) from the denominator.

1/sqrt(a) = sqrt(a)/a

•Multiply top and bottom by sqrt(2)
•= sqrt(2)/2
•Removes the surd from the denominator

Why learn this

It makes fractions easier to compute and compare - important in exams and in engineering maths.

💡 Memory trick

Multiply top and bottom by the same surd - the root moves upstairs.

MathematicsRational and irrational numbersmedium

Differentiate between rational and irrational numbers with an example each.

Reveal answer▾

What it is

A rational number can be written as p/q with integers p and q (q not 0); an irrational number cannot.

Answer

A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not zero; its decimal expansion is either terminating or non-terminating but recurring, for example 3/4 = 0.75 or 1/3 = 0.333... An irrational number cannot be written as p/q, and its decimal expansion is non-terminating and non-recurring, for example sqrt(2) = 1.41421... or pi.

Rational number = p/q, q not equal to 0

•Rational: p/q form, q not 0
•Rational decimals terminate or recur
•Irrational: cannot be written as p/q
•Irrational decimals are non-terminating, non-recurring (sqrt 2, pi)

Why learn this

Together they make up the real numbers used on the number line.

💡 Memory trick

Rational = ratio p/q (terminating or repeating decimal). Irrational = non-terminating, non-repeating.

MathematicsRationalising the denominatormedium

Rationalise the denominator of 1 / sqrt(3).

Reveal answer▾

What it is

Rationalising removes a surd from the denominator by multiplying by a suitable factor.

Answer

To rationalise 1/sqrt(3), multiply both the numerator and the denominator by sqrt(3): (1 x sqrt(3)) / (sqrt(3) x sqrt(3)) = sqrt(3) / 3. So 1/sqrt(3) = sqrt(3)/3, which now has a rational denominator.

1/sqrt(a) = sqrt(a)/a

•Multiply numerator and denominator by the surd
•sqrt(3) x sqrt(3) = 3
•1/sqrt(3) = sqrt(3)/3
•Use the conjugate for a + sqrt(b) type denominators

Why learn this

It simplifies expressions and makes them easier to compute.

💡 Memory trick

Multiply top and bottom by the surd (or its conjugate) to clear the root below.

MathematicsLaws of exponents for real numbersmedium

Simplify 2^(1/2) x 2^(1/2) and (8)^(2/3).

Reveal answer▾

What it is

The laws of exponents also apply to rational powers (roots) of positive real numbers.

Answer

Using a^m x a^n = a^(m+n): 2^(1/2) x 2^(1/2) = 2^(1/2 + 1/2) = 2^1 = 2. For (8)^(2/3), write 8 = 2^3, so (2^3)^(2/3) = 2^(3 x 2/3) = 2^2 = 4. So the answers are 2 and 4.

a^m x a^n = a^(m+n) ; (a^m)^n = a^(mn)

•a^m x a^n = a^(m+n)
•a^(1/n) = nth root of a
•2^(1/2) x 2^(1/2) = 2
•(8)^(2/3) = 4

Why learn this

They let us simplify expressions involving powers and roots.

💡 Memory trick

a^(1/n) means the nth root of a. Multiply powers -> add; divide -> subtract.

Polynomials5 questions

MathematicsRemainder theoremmedium

Using the remainder theorem, find the remainder when p(x) = x^3 + 3x^2 + 3x + 1 is divided by (x + 1).

Reveal answer▾

What it is

The remainder theorem says the remainder of p(x) divided by (x - a) is simply p(a).

Answer

By the remainder theorem, the remainder when p(x) is divided by (x + 1) is p(-1). p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0. Since the remainder is 0, (x + 1) is a factor of p(x).

Remainder theorem: remainder = p(a) when dividing by (x - a)

•Remainder = p(-1)
•p(-1) = -1 + 3 - 3 + 1 = 0
•Remainder 0 -> (x + 1) is a factor

Why learn this

It checks factors of big polynomials in one step - a shortcut used all through algebra.

💡 Memory trick

Dividing by (x + 1)? Just plug in x = -1. Set the bracket to zero, then substitute.

MathematicsDegree and types of polynomialseasy

Define the degree of a polynomial. Name the polynomials of degree 1, 2 and 3.

Reveal answer▾

What it is

The degree of a polynomial is the highest power of the variable; polynomials are named by their degree.

Answer

The degree of a polynomial is the highest power of the variable present in it. A polynomial of degree 1, such as 2x + 3, is called a linear polynomial; a polynomial of degree 2, such as x^2 + 5x + 6, is called a quadratic polynomial; and a polynomial of degree 3, such as x^3 - 2x + 1, is called a cubic polynomial. A constant like 7 has degree 0.

Degree = highest power of the variable

•Degree = highest power of the variable
•Degree 1: linear
•Degree 2: quadratic
•Degree 3: cubic

Why learn this

Knowing the type helps decide how to factorise and solve.

💡 Memory trick

Degree 1 = linear, 2 = quadratic, 3 = cubic.

MathematicsRemainder theoremmedium

Find the remainder when p(x) = x^3 + 3x^2 + 3x + 1 is divided by (x + 1).

Reveal answer▾

What it is

When a polynomial p(x) is divided by (x - a), the remainder is p(a).

Answer

By the remainder theorem, the remainder when p(x) is divided by (x - a) is p(a). Here the divisor is (x + 1) = (x - (-1)), so a = -1. Substitute: p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0. So the remainder is 0, which means (x + 1) is a factor.

Remainder of p(x) / (x - a) = p(a)

•Remainder theorem: remainder = p(a)
•Divisor (x + 1) means a = -1
•p(-1) = -1 + 3 - 3 + 1 = 0
•Remainder 0 means (x + 1) is a factor

Why learn this

It finds the remainder without doing long division.

💡 Memory trick

Remainder on dividing by (x - a) is just p(a) - substitute a into the polynomial.

MathematicsFactor theoremmedium

State the factor theorem. Is (x - 2) a factor of p(x) = x^2 - 5x + 6?

Reveal answer▾

What it is

(x - a) is a factor of a polynomial p(x) if and only if p(a) = 0.

Answer

The factor theorem states that (x - a) is a factor of a polynomial p(x) if and only if p(a) = 0. To test (x - 2), substitute a = 2: p(2) = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0. Since p(2) = 0, (x - 2) is a factor of x^2 - 5x + 6.

(x - a) is a factor of p(x) if p(a) = 0

•(x - a) is a factor if p(a) = 0
•Test x - 2: put a = 2
•p(2) = 4 - 10 + 6 = 0
•So (x - 2) is a factor

Why learn this

It quickly tests whether a given linear expression divides a polynomial exactly.

💡 Memory trick

If p(a) = 0, then (x - a) is a factor - a is a zero of the polynomial.

MathematicsAlgebraic identity for (a + b + c)^2medium

Expand (x + 2y + 3z)^2 using a standard identity.

Reveal answer▾

What it is

The identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca expands the square of a trinomial.

Answer

Using (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca with a = x, b = 2y and c = 3z: = x^2 + (2y)^2 + (3z)^2 + 2(x)(2y) + 2(2y)(3z) + 2(3z)(x) = x^2 + 4y^2 + 9z^2 + 4xy + 12yz + 6zx.

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

•(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca
•Square each term
•Add twice each pair product
•Answer: x^2 + 4y^2 + 9z^2 + 4xy + 12yz + 6zx

Why learn this

It speeds up expansion and appears in many algebra questions.

💡 Memory trick

Square each term, then add twice each pair product (ab, bc, ca).

Linear Equations in Two Variables3 questions

MathematicsIntercepts of a lineeasy

Where does the line x + y = 5 meet the x-axis and the y-axis?

Reveal answer▾

What it is

A straight line meets the axes at its intercepts - put y = 0 for the x-axis, x = 0 for the y-axis.

Answer

On the x-axis, y = 0, so x + 0 = 5 gives x = 5; the line meets the x-axis at (5, 0). On the y-axis, x = 0, so 0 + y = 5 gives y = 5; it meets the y-axis at (0, 5). These two intercept points are enough to draw the straight line.

Intercepts: set y = 0 for x-axis, x = 0 for y-axis

•x-axis: put y = 0 -> (5, 0)
•y-axis: put x = 0 -> (0, 5)
•Two points define the straight line

Why learn this

Two intercepts are the fastest way to draw any line - used in graphs, economics and coding.

💡 Memory trick

Zero the other one: y = 0 gives the x-cut, x = 0 gives the y-cut.

MathematicsEquation of a line (y = mx + c)medium

For the line y = 2x + 1, find y when x = 3. Slide m, x and c to explore.

Reveal answer▾

What it is

A straight line is described by y = mx + c, where m is the slope and c is the y-intercept.

y = m × x + c

Slope m2x value3Intercept c1

y7.0

Answer

y = m x + c = 2 x 3 + 1 = 6 + 1 = 7. The slope m tells how steeply y changes with x, and c is where the line crosses the y-axis (its value when x = 0).

y = m x + c

•y = m x + c
•m is the slope, c the y-intercept
•c is the value of y when x = 0

Why learn this

It's the workhorse equation for graphs, rates and trends.

💡 Memory trick

y = mx + c: slope times x, plus the intercept.

MathematicsSolutions of a linear equation in two variablesmedium

How many solutions does a linear equation in two variables have? Find two solutions of x + y = 5.

Reveal answer▾

What it is

A linear equation in two variables has infinitely many solutions, which lie on a straight line when graphed.

Answer

A linear equation in two variables of the form ax + by + c = 0 has infinitely many solutions, and all of them lie on a straight line when plotted on a graph. For x + y = 5, we can choose values of x and find y: if x = 0, then y = 5, giving (0, 5); if x = 2, then y = 3, giving (2, 3). So (0, 5) and (2, 3) are two of its infinitely many solutions.

ax + by + c = 0

•Form ax + by + c = 0
•Infinitely many solutions
•Graph is a straight line
•Solutions of x + y = 5: (0,5), (2,3), ...

Why learn this

It models relationships between two quantities, such as cost and number of items.

💡 Memory trick

Pick any x, solve for y - each pair (x, y) is one point on the line.

Heron's Formula4 questions

MathematicsArea of a trianglemedium

Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm using Heron's formula.

Reveal answer▾

What it is

Heron's formula finds a triangle's area from just its three sides, using the semi-perimeter.

Answer

First find the semi-perimeter s = (3 + 4 + 5)/2 = 6 cm. Heron's formula gives Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[6(6 - 3)(6 - 4)(6 - 5)] = sqrt[6 x 3 x 2 x 1] = sqrt[36] = 6 square cm.

Area = sqrt[s(s-a)(s-b)(s-c)], s = (a+b+c)/2

•s = (a + b + c)/2 = 6 cm
•Area = sqrt[s(s-a)(s-b)(s-c)]
•= sqrt[36] = 6 cm^2

Why learn this

It measures land, plots and any triangle where you cannot easily find the height.

💡 Memory trick

Half the perimeter is 's'; Area = sqrt of s times the three (s - side) gaps.

MathematicsArea by Heron's formulamedium

Find the area of a triangle with sides 3, 4 and 5 units using Heron's formula. Slide to explore.

Reveal answer▾

What it is

Heron's formula gives the area of a triangle from the lengths of its three sides.

Heron: √(s(s−a)(s−b)(s−c))

Side a3 unitsSide b4 unitsSide c5 units

Area6.00 units²

Answer

First the semi-perimeter s = (a+b+c)/2 = (3+4+5)/2 = 6. Then area = sqrt(s(s-a)(s-b)(s-c)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units. (This is a right triangle, so 1/2 x 3 x 4 = 6 agrees.)

area = sqrt(s(s-a)(s-b)(s-c))

•s = (a + b + c) / 2
•Area = sqrt(s(s-a)(s-b)(s-c))
•Uses only the three sides

Why learn this

It works even when the height is not known, using only the sides.

💡 Memory trick

s = (a+b+c)/2, then area = sqrt(s(s-a)(s-b)(s-c)).

MathematicsArea of an equilateral trianglemedium

Find the area of an equilateral triangle of side 6 units. Slide the side to explore.

Reveal answer▾

What it is

The area of an equilateral triangle is root three over four times the square of its side.

Area = (√3 ÷ 4) × a²

Side a6 units

Area15.59 units²

Answer

Area = (sqrt(3) / 4) x side^2 = (1.732 / 4) x 6^2 = 0.433 x 36 = 15.59 square units. This is a special case of Heron's formula when all three sides are equal.

A = (sqrt(3) / 4) x side^2

•Area = (sqrt(3) / 4) x side^2
•About 0.433 x side^2
•Special case of Heron's formula

Why learn this

It's a quick formula for the most symmetric triangle.

💡 Memory trick

Area = (sqrt(3) / 4) x side^2, about 0.433 x side^2.

MathematicsArea of a triangle by Heron's formulamedium

Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm using Heron's formula.

Reveal answer▾

What it is

Heron's formula finds the area of a triangle from the lengths of its three sides.

Answer

First find the semi-perimeter s = (a + b + c)/2 = (3 + 4 + 5)/2 = 12/2 = 6 cm. Then area = sqrt(s(s - a)(s - b)(s - c)) = sqrt(6(6 - 3)(6 - 4)(6 - 5)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 cm^2. So the area of the triangle is 6 square centimetres.

Area = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2

•s = (a + b + c)/2 = 6
•Area = sqrt(s(s-a)(s-b)(s-c))
•sqrt(6 x 3 x 2 x 1) = sqrt(36)
•Area = 6 cm^2

Why learn this

It works even when the height of the triangle is not known.

💡 Memory trick

Find s = (a+b+c)/2, then area = sqrt(s(s-a)(s-b)(s-c)).

Surface Areas and Volumes18 questions

MathematicsVolume of a spheremedium

Find the volume of a sphere of radius 7 cm. (Take pi = 22/7.)

Reveal answer▾

What it is

A sphere's volume grows with the cube of its radius: V = (4/3) pi r^3.

radius r = 7 cm

V = 4/3 π r³ ≈ 1437.3 cm³

Sphere — slide the radius; volume grows with r³ (π = 22/7)

Answer

Volume of a sphere = (4/3) x pi x r^3 = (4/3) x (22/7) x 7^3 = (4/3) x (22/7) x 343 = (4 x 22 x 49)/3 = 4312/3, which is about 1437.3 cubic cm.

V(sphere) = (4/3) pi r^3

•V = (4/3) pi r^3
•= (4/3)(22/7)(343)
•= 4312/3 = about 1437.3 cm^3

Why learn this

It measures balls, planets, bubbles and tanks - anywhere something is round.

💡 Memory trick

Four-thirds pi r-cubed: V = (4/3) pi r^3 - the roundest formula in maths.

MathematicsSurface area of a cubeeasy

Find the surface area of a cube of side 4 units. Slide the side to explore.

Reveal answer▾

What it is

The surface area of a cube is six times the area of one square face.

Cube surface area = 6 × side²

Side4 units

Surface area96 units²

Answer

Surface area = 6 x side^2 = 6 x 4^2 = 6 x 16 = 96 square units. A cube has six identical square faces, so we find the area of one face and multiply by six.

S = 6 x side^2

•Surface area = 6 x side^2
•A cube has 6 equal faces
•Unit: square units

Why learn this

It tells how much material wraps or paints a cubic box.

💡 Memory trick

Surface area = 6 x side^2 - six equal faces.

MathematicsVolume of a conemedium

Find the volume of a cone of radius 3 and height 7 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The volume of a cone is one third of pi times the square of the radius times the height.

Cone volume = ⅓ × π × r² × h

Radius r3 unitsHeight h7 units

Volume66.0 units³

Answer

Volume = 1/3 x pi x r^2 x h = 1/3 x 3.14159 x 3^2 x 7 = 1/3 x 3.14159 x 9 x 7 = 65.97 cubic units. A cone holds exactly one third of the cylinder with the same base and height.

V = (1/3) x pi x r^2 x h

•Volume = 1/3 x pi x r^2 x h
•One third of the matching cylinder
•Unit: cubic units

Why learn this

It measures the space in cones, funnels and ice-cream cones.

💡 Memory trick

Volume = 1/3 x pi x r^2 x h - a third of the cylinder.

MathematicsSurface area of a cylindermedium

Find the total surface area of a cylinder of radius 3 and height 7 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The total surface area of a cylinder is the curved side plus its two circular ends.

Cylinder surface = 2πr(r + h)

Radius r3 unitsHeight h7 units

Total surface area188.5 units²

Answer

Total surface area = 2 pi r (r + h) = 2 x 3.14159 x 3 x (3 + 7) = 2 x 3.14159 x 3 x 10 = 188.5 square units. This is the curved surface 2 pi r h plus the two circular ends 2 pi r^2.

S = 2 pi r (r + h)

•Total surface = 2 pi r (r + h)
•Curved side (2 pi r h) + two ends (2 pi r^2)
•Unit: square units

Why learn this

It's the metal needed to make a closed tin or tank.

💡 Memory trick

Total surface = 2 pi r (r + h).

MathematicsSurface area of a cuboidmedium

Find the surface area of a cuboid 3 x 4 x 5 units. Slide the dimensions to explore.

Reveal answer▾

What it is

The surface area of a cuboid is twice the sum of the areas of its three different faces.

Cuboid surface = 2(lb + bh + hl)

Length3 unitsBreadth4 unitsHeight5 units

Surface area94 units²

Answer

Surface area = 2(lb + bh + hl) = 2(3x4 + 4x5 + 5x3) = 2(12 + 20 + 15) = 2 x 47 = 94 square units. A cuboid has three pairs of identical faces, so we add the three distinct faces and double the total.

S = 2(lb + bh + hl)

•Surface = 2(lb + bh + hl)
•Three pairs of identical faces
•Unit: square units

Why learn this

It's the card needed to make a closed rectangular box.

💡 Memory trick

Surface = 2(lb + bh + hl).

MathematicsVolume of a hemispheremedium

Find the volume of a hemisphere of radius 6 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The volume of a hemisphere is two thirds of pi times the cube of the radius.

Hemisphere volume = ⅔ × π × r³

Radius r6 units

Volume452.4 units³

Answer

Volume = 2/3 x pi x r^3 = 2/3 x 3.14159 x 6^3 = 2/3 x 3.14159 x 216 = 452.39 cubic units. A hemisphere is half a sphere, so its volume is half of 4/3 x pi x r^3, which is 2/3 x pi x r^3.

V = (2/3) x pi x r^3

•Volume = 2/3 x pi x r^3
•Half of a full sphere's volume
•Unit: cubic units

Why learn this

It sizes domes, bowls and half-spherical tanks.

💡 Memory trick

Volume = 2/3 x pi x r^3 - half a sphere.

MathematicsSlant height of a conemedium

A cone has radius 3 and height 4 units. Find its slant height. Slide to explore.

Reveal answer▾

What it is

The slant height of a cone is found from its radius and vertical height using the Pythagoras theorem.

Slant height l = √(r² + h²)

Radius r3 unitsHeight h4 units

Slant height l5.00 units

Answer

Slant height l = sqrt(r^2 + h^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. The radius, the vertical height and the slant height form a right triangle, so Pythagoras gives the slant.

l = sqrt(r^2 + h^2)

•l = sqrt(r^2 + h^2)
•From the Pythagoras theorem
•Used for the curved surface area

Why learn this

It's needed to work out the curved surface area of a cone.

💡 Memory trick

l = sqrt(r^2 + h^2) - the hypotenuse of r and h.

MathematicsCurved surface area of a conemedium

Find the curved surface area of a cone with radius 3 and slant height 5 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The curved surface area of a cone is pi times the radius times the slant height.

Cone curved area = π × r × l

Radius r3 unitsSlant height l5 units

Curved surface area47.1 units²

Answer

Curved surface area = pi x r x l = 3.14159 x 3 x 5 = 47.12 square units. Note that l is the slant height along the sloping side, not the vertical height of the cone.

CSA = pi x r x l

•Curved area = pi x r x l
•l is the slant height
•Unit: square units

Why learn this

It's the paper needed to make the cone's sloping side, like an ice-cream wrapper.

💡 Memory trick

Curved area = pi x r x l (l is the slant height).

MathematicsTotal surface area of a conemedium

Find the total surface area of a cone with radius 3 and slant height 5 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The total surface area of a cone is its curved surface plus its circular base.

Cone total area = π × r × (l + r)

Radius r3 unitsSlant height l5 units

Total surface area75.4 units²

Answer

Total surface area = pi x r x (l + r) = 3.14159 x 3 x (5 + 3) = 3.14159 x 3 x 8 = 75.40 square units. This is the curved surface (pi x r x l) plus the base circle (pi x r^2).

TSA = pi x r x (l + r)

•Total area = pi x r x (l + r)
•Curved surface + base circle
•Unit: square units

Why learn this

It's the material to make a closed cone, like a party hat with a base.

💡 Memory trick

Total area = pi x r x (l + r).

MathematicsCurved surface area of a hemisphereeasy

Find the curved surface area of a hemisphere of radius 6 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The curved surface area of a hemisphere is two pi times the square of the radius.

Hemisphere curved area = 2 × π × r²

Radius r6 units

Curved surface area226.2 units²

Answer

Curved surface area = 2 x pi x r^2 = 2 x 3.14159 x 6^2 = 2 x 3.14159 x 36 = 226.19 square units. A full sphere has surface 4 x pi x r^2, so the curved part of a hemisphere is exactly half of that.

CSA = 2 x pi x r^2

•Curved area = 2 x pi x r^2
•Half a sphere's surface (4 pi r^2)
•Unit: square units

Why learn this

It's the dome surface of bowls, domes and half-spheres.

💡 Memory trick

Curved area = 2 x pi x r^2 - half a sphere's surface.

MathematicsDiagonal of a cuboidmedium

Find the longest diagonal of a cuboid 3 x 4 x 12 units. Slide the dimensions to explore.

Reveal answer▾

What it is

The longest diagonal of a cuboid is the square root of the sum of the squares of its three dimensions.

Cuboid diagonal = √(l² + b² + h²)

Length3 unitsBreadth4 unitsHeight12 units

Space diagonal13.00 units

Answer

Space diagonal = sqrt(l^2 + b^2 + h^2) = sqrt(3^2 + 4^2 + 12^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13 units. It extends the Pythagoras theorem into three dimensions.

diagonal = sqrt(l^2 + b^2 + h^2)

•Diagonal = sqrt(l^2 + b^2 + h^2)
•Pythagoras in 3D
•Longest rod that fits inside

Why learn this

It's the longest straight rod or stick that can fit inside a box.

💡 Memory trick

Diagonal = sqrt(l^2 + b^2 + h^2).

MathematicsCurved surface area of a cylindereasy

Find the curved surface area of a cylinder with radius 3 and height 7 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The curved surface area of a cylinder is two pi times the radius times the height.

Cylinder curved area = 2 × π × r × h

Radius r3 unitsHeight h7 units

Curved surface area131.9 units²

Answer

Curved surface area = 2 x pi x r x h = 2 x 3.14159 x 3 x 7 = 131.95 square units. It is the rectangle you would get by unrolling the curved side, with width equal to the circumference and height equal to the cylinder's height.

CSA = 2 x pi x r x h

•Curved area = 2 x pi x r x h
•The unrolled side is a rectangle
•Excludes the two circular ends

Why learn this

It's the label that wraps around a tin, without the top and bottom.

💡 Memory trick

Curved area = 2 x pi x r x h.

MathematicsDiagonal of a cubemedium

Find the longest diagonal of a cube of side 5 units. Slide the side to explore.

Reveal answer▾

What it is

The longest diagonal of a cube is its side length times the square root of three.

Cube diagonal = side × √3

Side5 units

Space diagonal8.66 units

Answer

Space diagonal = side x sqrt(3) = 5 x 1.732 = 8.66 units. It comes from applying Pythagoras twice - once across a face, then through the solid - giving sqrt(side^2 + side^2 + side^2) = side x sqrt(3).

diagonal = side x sqrt(3)

•Diagonal = side x sqrt(3)
•About side x 1.732
•Pythagoras in three dimensions

Why learn this

It's the longest straight rod that fits inside a cubical box.

💡 Memory trick

Diagonal = side x sqrt(3), about side x 1.732.

MathematicsTotal surface area of a hemispheremedium

Find the total surface area of a solid hemisphere of radius 6 units (pi = 3.14159). Slide to explore.

Reveal answer▾

What it is

The total surface area of a solid hemisphere is three pi times the square of the radius.

Hemisphere total area = 3 × π × r²

Radius r6 units

Total surface area339.3 units²

Answer

Total surface area = 3 x pi x r^2 = 3 x 3.14159 x 6^2 = 3 x 3.14159 x 36 = 339.29 square units. It is the curved dome (2 pi r^2) plus the flat circular base (pi r^2), which together make 3 pi r^2.

TSA = 3 x pi x r^2

•Total = 3 x pi x r^2
•Curved 2 pi r^2 + base pi r^2
•Unit: square units

Why learn this

It's the full surface of a solid dome, including its flat base.

💡 Memory trick

Total = 3 x pi x r^2 (curved 2 pi r^2 + base pi r^2).

MathematicsEdge of a cube from volumemedium

A cube has a volume of 64 cubic units. Find its edge. Slide the volume to explore.

Reveal answer▾

What it is

The edge of a cube is the cube root of its volume.

Edge = cube root of volume

Volume64 units³

Edge length4.00 units

Answer

Edge = cube root of volume = cube root of 64 = 4 units, because 4 x 4 x 4 = 64. It simply reverses the volume formula V = side^3.

side = cube root of volume

•Edge = cube root of volume
•Reverses V = side^3
•Volume 64 -> edge 4

Why learn this

It reverses the volume formula to find the side.

💡 Memory trick

Since V = side^3, side = cube root of V.

MathematicsCurved surface area and volume of a conemedium

Find the volume of a cone of radius 3 cm and height 7 cm. (Take pi = 22/7.)

Reveal answer▾

What it is

A cone has curved surface area pi r l and volume (1/3) pi r^2 h.

Answer

Volume of a cone = (1/3) pi r^2 h = (1/3) x (22/7) x 3^2 x 7 = (1/3) x (22/7) x 9 x 7 = (1/3) x 22 x 9 = (1/3) x 198 = 66 cm^3. So the volume of the cone is 66 cubic centimetres.

Volume of cone = (1/3) pi r^2 h ; CSA = pi r l

•Volume of cone = (1/3) pi r^2 h
•= (1/3)(22/7)(9)(7)
•= 66 cm^3
•Slant height l = sqrt(r^2 + h^2)

Why learn this

It is used for ice-cream cones, funnels and conical tents.

💡 Memory trick

Cone volume is one-third of a cylinder of the same base and height.

MathematicsSurface area and volume of a spheremedium

Find the surface area of a sphere of radius 7 cm. (Take pi = 22/7.)

Reveal answer▾

What it is

A sphere has surface area 4 pi r^2 and volume (4/3) pi r^3.

Answer

Surface area of a sphere = 4 pi r^2 = 4 x (22/7) x 7^2 = 4 x (22/7) x 49 = 4 x 22 x 7 = 616 cm^2. So the surface area of the sphere is 616 square centimetres.

Surface area = 4 pi r^2 ; Volume = (4/3) pi r^3

•Surface area of sphere = 4 pi r^2
•= 4 x (22/7) x 49
•= 616 cm^2
•Volume of sphere = (4/3) pi r^3

Why learn this

It is used for balls, globes and bubbles.

💡 Memory trick

Sphere surface = 4 pi r^2; volume = (4/3) pi r^3.

MathematicsTotal surface area of a cylindermedium

Find the total surface area of a closed cylinder of radius 7 cm and height 10 cm. (Take pi = 22/7.)

Reveal answer▾

What it is

A cylinder has curved surface area 2 pi r h and total surface area 2 pi r (r + h).

Answer

Total surface area of a closed cylinder = 2 pi r (r + h) = 2 x (22/7) x 7 x (7 + 10) = 2 x 22 x 17 = 748 cm^2. So the total surface area of the cylinder is 748 square centimetres.

TSA of cylinder = 2 pi r (r + h)

•CSA = 2 pi r h
•TSA = 2 pi r (r + h)
•= 2 x (22/7) x 7 x 17
•TSA = 748 cm^2

Why learn this

It is used for pipes, tanks, tins and pillars.

💡 Memory trick

TSA of a closed cylinder = 2 pi r (h + r) (curved surface plus two circles).

Statistics6 questions

MathematicsMean of dataeasy

Find the mean of the data: 10, 15, 20, 25, 30.

Reveal answer▾

What it is

The mean (average) is the total of all values divided by how many values there are.

Answer

Mean = (sum of all observations) / (number of observations) = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20.

Mean = (sum of observations) / (number of observations)

•Mean = sum of values / number of values
•Sum = 100, count = 5
•Mean = 20

Why learn this

It summarises data in one number - used in marks, cricket averages, economics and science.

💡 Memory trick

Add them all up, divide by how many. Mean = sum / count.

MathematicsMean (average)easy

Five values add up to 250. Find their mean. Slide the values to explore.

Reveal answer▾

What it is

The mean of a set of values is their sum divided by how many values there are.

Mean = sum ÷ number of items

Sum of values250Number of items5

Mean (average)50.0

Answer

Mean = sum of values / number of values = 250 / 5 = 50. The mean is the value each item would have if the total were shared out equally among them.

mean = sum of values / number of values

•Mean = sum / number of items
•Also called the average
•Shares the total equally

Why learn this

It's the everyday 'average' used for marks, scores and data.

💡 Memory trick

Mean = sum / count. Share the total equally.

MathematicsRange of dataeasy

A data set has a highest value of 80 and a lowest of 20. Find the range. Slide to explore.

Reveal answer▾

What it is

The range of a set of data is the difference between its highest and lowest values.

Range = highest − lowest

Highest value80Lowest value20

Range60

Answer

Range = highest value - lowest value = 80 - 20 = 60. A larger range means the data is more spread out; a small range means the values are close together.

range = highest value - lowest value

•Range = highest - lowest
•Simplest measure of spread
•Larger range -> more spread out

Why learn this

It's the quickest way to describe how spread out data is.

💡 Memory trick

Range = highest - lowest.

MathematicsClass markeasy

Find the class mark of the interval 10 to 20. Slide the limits to explore.

Reveal answer▾

What it is

The class mark is the midpoint of a class interval, the average of its lower and upper limits.

Class mark = (lower + upper) ÷ 2

Lower limit10Upper limit20

Class mark15.0

Answer

Class mark = (lower limit + upper limit) / 2 = (10 + 20) / 2 = 15. It is the midpoint of the interval and stands in for every value in that class when we calculate a grouped mean.

class mark = (lower + upper) / 2

•Class mark = (lower + upper) / 2
•Midpoint of the interval
•Used for grouped-data mean

Why learn this

It represents a whole interval when finding the mean of grouped data.

💡 Memory trick

Class mark = (lower + upper) / 2.

MathematicsMean of ungrouped dataeasy

Find the mean of the data: 10, 12, 14, 16, 18.

Reveal answer▾

What it is

The mean (average) is the sum of all observations divided by the number of observations.

Answer

The mean is the sum of all observations divided by the number of observations. Sum = 10 + 12 + 14 + 16 + 18 = 70. Number of observations = 5. Mean = 70 / 5 = 14. So the mean of the data is 14.

Mean = (sum of observations) / (number of observations)

•Mean = sum / number of observations
•Sum = 70
•Number = 5
•Mean = 14

Why learn this

It is the most common way to represent a set of data by a single value.

💡 Memory trick

Mean = sum of values / number of values.

MathematicsMedian and modemedium

Find the median and mode of the data: 4, 6, 6, 8, 10.

Reveal answer▾

What it is

The median is the middle value of ordered data; the mode is the most frequently occurring value.

Answer

The data is already arranged in ascending order: 4, 6, 6, 8, 10. There are 5 observations (odd number), so the median is the middle (3rd) value, which is 6. The mode is the value that occurs most often; here 6 occurs twice while all others occur once, so the mode is 6. Thus both the median and the mode are 6.

Median = middle value (for odd n) ; Mode = most frequent value

•Arrange data in order first
•Median = middle value (odd count)
•Median = 6
•Mode = most frequent value = 6

Why learn this

The median resists extreme values and the mode shows the most common item.

💡 Memory trick

Median = MIDdle of sorted data; Mode = MOST frequent value.

Triangles5 questions

MathematicsPythagoras theoremmedium

The two shorter sides of a right triangle are 3 and 4. Find the hypotenuse. Slide to explore.

Reveal answer▾

What it is

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Hypotenuse c = √(a² + b²)

Side a3Side b4

Hypotenuse c5.00 units

Answer

By the Pythagoras theorem, c^2 = a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = sqrt(25) = 5. The hypotenuse is always the longest side and lies opposite the right angle.

c = sqrt(a^2 + b^2)

•c^2 = a^2 + b^2
•c is the hypotenuse (longest side)
•3-4-5 is the classic example

Why learn this

It measures distances we cannot walk in a straight line - used in construction, navigation and graphics.

💡 Memory trick

c^2 = a^2 + b^2. The 3-4-5 triangle is the classic check.

MathematicsExterior angle theoremeasy

Two interior angles of a triangle are 60 and 70 degrees. Find the exterior angle at the third vertex. Slide to explore.

Reveal answer▾

What it is

An exterior angle of a triangle equals the sum of the two interior angles opposite to it.

Exterior angle = angle 1 + angle 2

Interior angle 160 °Interior angle 270 °

Exterior angle130 °

Answer

Exterior angle = angle 1 + angle 2 = 60 + 70 = 130 degrees. This works because the exterior angle and the third interior angle sit on a straight line (180 degrees), and the three interior angles also add to 180 degrees.

exterior angle = interior angle 1 + interior angle 2

•Exterior angle = sum of two remote interior angles
•Equals 180 minus the adjacent interior angle
•Follows from the angle sum of a triangle

Why learn this

It's a quick way to find an unknown angle without using the full 180 degree rule.

💡 Memory trick

Exterior angle = sum of the two remote interior angles.

MathematicsAngle sum and exterior angle propertyeasy

Two interior angles of a triangle are 50 and 60 degrees. Find the third angle and the exterior angle at the third vertex.

Reveal answer▾

What it is

The interior angles of a triangle sum to 180 degrees, and an exterior angle equals the sum of the two opposite interior angles.

Answer

By the angle sum property, the three interior angles add to 180 degrees, so the third angle = 180 - (50 + 60) = 180 - 110 = 70 degrees. By the exterior angle property, the exterior angle at the third vertex equals the sum of the two opposite interior angles = 50 + 60 = 110 degrees (which is also 180 - 70).

Angle sum = 180 degrees ; exterior angle = sum of two opposite interior angles

•Interior angles of a triangle sum to 180 degrees
•Third angle = 180 - 110 = 70 degrees
•Exterior angle = sum of opposite interior angles
•Exterior angle = 50 + 60 = 110 degrees

Why learn this

These are the most-used rules for finding angles in triangles.

💡 Memory trick

Exterior angle = sum of the two remote (opposite) interior angles.

MathematicsCriteria for congruence of trianglesmedium

State the criteria for the congruence of two triangles.

Reveal answer▾

What it is

Two triangles are congruent if they satisfy SSS, SAS, ASA, AAS or RHS.

Answer

Two triangles are congruent (equal in all respects) if any one of the following holds: SSS, when the three sides of one equal the three sides of the other; SAS, when two sides and the included angle are equal; ASA, when two angles and the included side are equal; AAS, when two angles and a non-included side are equal; and RHS, when in two right triangles the hypotenuse and one side are equal. Note that there is no SSA/ASS congruence rule.

Congruence: SSS, SAS, ASA, AAS, RHS

•SSS: three sides equal
•SAS: two sides and included angle
•ASA / AAS: two angles and a side
•RHS: right angle, hypotenuse and one side

Why learn this

Congruence lets us prove sides and angles equal in geometry.

💡 Memory trick

SSS, SAS, ASA, AAS, RHS - note there is NO ASS/SSA rule.

MathematicsProperties of an isosceles trianglemedium

State the two key properties of an isosceles triangle.

Reveal answer▾

What it is

In an isosceles triangle the angles opposite the equal sides are equal, and vice versa.

Answer

An isosceles triangle has two sides equal. Its key properties are: the angles opposite the two equal sides are equal (the base angles are equal); and conversely, if two angles of a triangle are equal, then the sides opposite them are equal, so the triangle is isosceles. Also, the perpendicular drawn from the vertex angle to the base bisects the base and the vertex angle.

Angles opposite equal sides are equal

•Two sides are equal
•Angles opposite equal sides are equal
•Equal angles imply equal opposite sides
•Vertex perpendicular bisects the base

Why learn this

It is a frequently used result in geometry proofs and angle problems.

💡 Memory trick

Equal sides -> equal base angles (and the reverse is also true).

Mensuration3 questions

MathematicsDiagonal of a squareeasy

Find the diagonal of a square of side 10 units. Slide the side to explore.

Reveal answer▾

What it is

The diagonal of a square is its side length times the square root of two.

Square diagonal = side × √2

Side10 units

Diagonal14.14 units

Answer

Diagonal = side x sqrt(2) = 10 x 1.414 = 14.14 units. This comes from the Pythagoras theorem on the two equal sides of the square: diagonal^2 = side^2 + side^2 = 2 x side^2.

diagonal = side x sqrt(2)

•Diagonal = side x sqrt(2)
•sqrt(2) is about 1.414
•From Pythagoras on two equal sides

Why learn this

It's the straight-line distance across a square tile or field.

💡 Memory trick

Diagonal = side x sqrt(2), about side x 1.414.

MathematicsDiagonal of a rectangleeasy

Find the diagonal of a rectangle 3 by 4 units. Slide the sides to explore.

Reveal answer▾

What it is

The diagonal of a rectangle is the square root of the sum of the squares of its length and breadth.

Rectangle diagonal = √(l² + b²)

Length3 unitsBreadth4 units

Diagonal5.00 units

Answer

Diagonal = sqrt(l^2 + b^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. The length, breadth and diagonal form a right triangle, so the Pythagoras theorem gives the diagonal.

diagonal = sqrt(l^2 + b^2)

•Diagonal = sqrt(l^2 + b^2)
•From the Pythagoras theorem
•3-4-5 is the classic case

Why learn this

It's the straight distance across a rectangular screen, room or field.

💡 Memory trick

Diagonal = sqrt(l^2 + b^2) - Pythagoras on the two sides.

MathematicsHeight of a triangle from areaeasy

A triangle has an area of 40 square units and a base of 10. Find its height. Slide to explore.

Reveal answer▾

What it is

The height of a triangle is twice its area divided by the base.

Height = 2 × area ÷ base

Area40 units²Base10 units

Height8.0 units

Answer

Height = 2 x area / base = 2 x 40 / 10 = 80 / 10 = 8 units. This is the area formula area = 1/2 x base x height rearranged to make the height the subject.

height = 2 x area / base

•Height = 2 x area / base
•Reverses area = 1/2 x base x height
•Area 40, base 10 -> height 8

Why learn this

It reverses the area formula to find a missing height.

💡 Memory trick

From area = 1/2 x base x height, height = 2 x area / base.

Lines and Angles3 questions

MathematicsSupplementary angleseasy

What is the supplement of a 60 degree angle? Slide the angle to explore.

Reveal answer▾

What it is

Two angles are supplementary when they add up to 180 degrees.

Supplement = 180° − angle

Angle60 °

Supplementary angle120 °

Answer

Supplement = 180 - angle = 180 - 60 = 120 degrees. Supplementary angles add to a straight angle (180 degrees), so the angles on one side of a straight line always sum to 180.

supplement = 180 - angle

•Supplement = 180 - angle
•The two add to 180 degrees
•Angles on a straight line are supplementary

Why learn this

Angles on a straight line are supplementary, a fact used throughout geometry.

💡 Memory trick

Supplement = 180 - angle.

MathematicsPairs of angleseasy

Define a linear pair and vertically opposite angles.

Reveal answer▾

What it is

When lines meet, they form angle pairs such as linear pairs (sum 180) and vertically opposite angles (equal).

Answer

A linear pair is a pair of adjacent angles formed when two lines intersect (or a ray stands on a line) such that their non-common arms form a straight line; the sum of a linear pair is 180 degrees. Vertically opposite angles are the pairs of opposite angles formed when two lines intersect; vertically opposite angles are always equal. For example, if two lines cross and one angle is 70 degrees, the angle opposite it is also 70 degrees, and its linear pair is 110 degrees.

Linear pair sum = 180 degrees

•Linear pair: adjacent angles summing to 180 degrees
•Vertically opposite angles are equal
•Formed when two lines intersect
•Used to find unknown angles

Why learn this

These relationships let us find unknown angles in figures.

💡 Memory trick

Linear pair sums to 180 degrees; vertically opposite angles are equal.

MathematicsParallel lines and a transversalmedium

State the angle relationships when a transversal cuts two parallel lines.

Reveal answer▾

What it is

A transversal cutting two parallel lines makes equal corresponding angles, equal alternate angles and co-interior angles that sum to 180.

Answer

When a transversal intersects two parallel lines, the following angle relationships hold: corresponding angles are equal; alternate interior angles are equal; alternate exterior angles are equal; and co-interior (allied or same-side interior) angles are supplementary, that is, they add up to 180 degrees. These properties can be used both to find unknown angles and to prove that two lines are parallel.

Co-interior angles sum = 180 degrees

•Corresponding angles are equal
•Alternate interior angles are equal
•Co-interior angles sum to 180 degrees
•Used to prove lines are parallel

Why learn this

These angle rules are used to prove lines parallel and to find angles.

💡 Memory trick

Corresponding equal, Alternate equal, Co-interior sum to 180 (C-shape adds to 180).

Coordinate Geometry1 question

MathematicsCartesian plane and quadrantseasy

In which quadrant do the points (3, -4) and (-2, 5) lie?

Reveal answer▾

What it is

The Cartesian plane is formed by two perpendicular axes and is divided into four quadrants.

Answer

On the Cartesian plane, the x-axis and y-axis meet at the origin and divide the plane into four quadrants. In a point (x, y): the first quadrant has both coordinates positive, the second has x negative and y positive, the third has both negative, and the fourth has x positive and y negative. The point (3, -4) has x positive and y negative, so it lies in the fourth quadrant. The point (-2, 5) has x negative and y positive, so it lies in the second quadrant.

A point is written as (x, y)

•Two perpendicular axes meet at the origin
•Quadrant signs: I(+,+), II(-,+), III(-,-), IV(+,-)
•(3, -4) is in quadrant IV
•(-2, 5) is in quadrant II

Why learn this

It lets us locate any point using an ordered pair of coordinates.

💡 Memory trick

Signs by quadrant: I (+,+), II (-,+), III (-,-), IV (+,-), going anticlockwise.

Quadrilaterals2 questions

MathematicsProperties of a parallelogrammedium

State any three properties of a parallelogram.

Reveal answer▾

What it is

In a parallelogram opposite sides and opposite angles are equal, and the diagonals bisect each other.

Answer

Three properties of a parallelogram are: opposite sides are equal and parallel; opposite angles are equal; and the diagonals bisect each other. A further property is that each diagonal divides the parallelogram into two congruent triangles, and any pair of adjacent angles is supplementary (sums to 180 degrees).

Diagonals of a parallelogram bisect each other

•Opposite sides equal and parallel
•Opposite angles equal
•Diagonals bisect each other
•A diagonal makes two congruent triangles

Why learn this

These properties are used to prove figures are parallelograms and to solve problems.

💡 Memory trick

Opposite sides equal, opposite angles equal, diagonals bisect each other.

MathematicsMidpoint theoremmedium

State the midpoint theorem. If the third side of a triangle is 10 cm, how long is the segment joining the midpoints of the other two sides?

Reveal answer▾

What it is

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Answer

The midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is equal to half of it. If the third side is 10 cm, then the segment joining the midpoints of the other two sides is parallel to it and its length is 10 / 2 = 5 cm.

Midsegment = (1/2) x third side

•Joins midpoints of two sides
•Parallel to the third side
•Length = half of the third side
•Half of 10 cm = 5 cm

Why learn this

It is used to prove parallelism and to find lengths in triangles.

💡 Memory trick

Join two midpoints -> the segment is parallel to and half the third side.

Circles2 questions

MathematicsChords and the perpendicular from the centremedium

State two properties relating the centre of a circle to its chords.

Reveal answer▾

What it is

The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

Answer

Two important chord properties are: the perpendicular drawn from the centre of a circle to a chord bisects the chord (and conversely, the line from the centre to the midpoint of a chord is perpendicular to it); and equal chords of a circle are equidistant from the centre (and chords that are equidistant from the centre are equal). These follow from the fact that all radii of a circle are equal.

Perpendicular from centre bisects the chord

•Perpendicular from centre bisects a chord
•Line from centre to a chord's midpoint is perpendicular
•Equal chords are equidistant from the centre
•Equidistant chords are equal

Why learn this

These chord properties are used to find lengths and prove results in circles.

💡 Memory trick

Perpendicular from the centre cuts a chord in half; equal chords are equally far from the centre.

MathematicsAngles and cyclic quadrilateralshard

State the angle-at-centre theorem and the property of a cyclic quadrilateral.

Reveal answer▾

What it is

The angle at the centre is twice the angle at the circumference on the same arc, and opposite angles of a cyclic quadrilateral sum to 180 degrees.

Answer

The angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference. A consequence is that angles in the same segment are equal, and the angle in a semicircle is a right angle. In a cyclic quadrilateral (one whose four vertices lie on a circle), the sum of each pair of opposite angles is 180 degrees, that is, the opposite angles are supplementary.

Angle at centre = 2 x angle at circumference

•Angle at centre = 2 x angle at circumference (same arc)
•Angles in the same segment are equal
•Angle in a semicircle = 90 degrees
•Cyclic quadrilateral: opposite angles sum to 180 degrees

Why learn this

These theorems solve a large number of circle geometry problems.

💡 Memory trick

Centre angle = 2 x circumference angle; cyclic quadrilateral opposite angles = 180.

Probability1 question

MathematicsEmpirical probabilityeasy

A coin is tossed 100 times and a head appears 60 times. Find the empirical probability of getting a head.

Reveal answer▾

What it is

Empirical (experimental) probability is the ratio of the number of times an event happens to the total number of trials.

Answer

Empirical probability of an event = (number of times the event occurs) / (total number of trials). Here a head appeared 60 times out of 100 tosses, so P(head) = 60/100 = 0.6. The probability of any event lies between 0 and 1, and the probabilities of all outcomes add up to 1.

P(E) = number of favourable trials / total trials

•P(event) = favourable outcomes / total trials
•P(head) = 60/100
•= 0.6
•Probability lies between 0 and 1

Why learn this

It estimates chances from real observed data, like coin tosses.

💡 Memory trick

P(event) = favourable outcomes / total trials, a value between 0 and 1.

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