For Class 8, 9 & 10
Master the basics - and everything after gets easier
Concept-first questions with clear model answers in Physics, Chemistry, Maths and Biology, all NCERT-aligned. Start early, build the habit, and walk into your boards, NEET and JEE already ahead.
Interactive lessons
learn by playingDrag, slide and build - watch each concept come alive, then reveal the answer.
200 interactive lessons
Ohm's law
Class 10 Physics
Slide V & R, watch the bulb glow
Open →pH scale
Class 10 Chemistry
Slide across acids and bases
Open →Atomic number and mass number
Class 9 Chemistry
Add protons & neutrons, build shells
Open →Laws of reflection
Class 8 Physics
Change the angle, watch it bounce
Open →Volume of a sphere
Class 9 Maths
Grow the radius, see the volume
Open →Area of a trapezium
Class 8 Maths
Drag the sides, read the area
Open →Power of a lens
Class 10 Physics
Move the object, trace the rays
Open →Food chain and energy flow
Class 10 Biology
Follow the energy as it flows
Open →Speed
Class 8 Physics
Slide distance & time, watch the speed
Open →Density
Class 9 Physics
Pack mass into volume, float or sink
Open →Work done
Class 9 Physics
Push harder or farther, watch work grow
Open →Kinetic energy
Class 9 Physics
Speed it up - energy grows with the square
Open →Power of a lens
Class 10 Physics
Shorten the focal length, boost the power
Open →Mole concept
Class 9 Chemistry
Weigh out grams, count the moles
Open →Avogadro's number
Class 9 Chemistry
Add moles, count the particles
Open →Microscope magnification
Class 8 Biology
Grow the image, read the magnification
Open →Population density
Class 10 Biology
Add individuals, shrink the land, see crowding
Open →Simple interest
Class 8 Maths
Slide money, rate & time, watch interest
Open →Pythagoras theorem
Class 9 Maths
Stretch the two sides, get the hypotenuse
Open →Probability of an event
Class 10 Maths
Change the outcomes, watch the odds
Open →Newton's second law
Class 9 Physics
Push a mass, pick an acceleration
Open →Momentum
Class 9 Physics
Slide mass & velocity, build momentum
Open →Pressure
Class 8 Physics
Shrink the area, feel the pressure rise
Open →Weight
Class 9 Physics
Change the planet's gravity, watch your weight
Open →Refractive index
Class 10 Physics
Slow light in the medium, raise the index
Open →Resistors in series
Class 10 Physics
Add two resistors in a line
Open →Mass percentage of a solution
Class 9 Chemistry
Dissolve solute, read the strength
Open →Concentration of a solution
Class 9 Chemistry
Pack solute into less liquid
Open →Population change
Class 10 Biology
Balance births against deaths
Open →Compound microscope
Class 8 Biology
Combine eyepiece & objective lenses
Open →Area of a circle
Class 8 Maths
Grow the radius, watch the area square
Open →Volume of a cuboid
Class 8 Maths
Stretch length, breadth & height
Open →Electronic configuration and valency
Class 9 Chemistry
Slide the atomic number, build the atom
Open →Homologous series (alkanes)
Class 10 Chemistry
Add carbons, name the compound
Open →Mass number
Class 9 Chemistry
Add protons & neutrons, get the mass number
Open →Power
Class 9 Physics
More work in less time = more power
Open →Potential energy
Class 9 Physics
Lift a mass higher, store energy
Open →Wave speed
Class 9 Physics
Tune frequency & wavelength, set the speed
Open →Electric current
Class 10 Physics
Push charge per second, get the current
Open →Percentage
Class 8 Maths
Compare part to whole as a %
Open →Electron dot structure
Class 9 Chemistry
Draw valence electrons as dots
Open →Acceleration
Class 9 Physics
Speed up over time, find acceleration
Open →Distance, speed and time
Class 8 Physics
Set speed & time, cover the distance
Open →Frequency and time period
Class 9 Physics
Shorten the period, raise the frequency
Open →Heating effect of current
Class 10 Physics
Raise the current, watch heating soar
Open →Area of a triangle
Class 8 Maths
Set base & height, halve the rectangle
Open →Area of a rectangle
Class 8 Maths
Set length & breadth, fill the area
Open →Mean (average)
Class 9 Maths
Share the total equally across items
Open →Discount
Class 8 Maths
Slide price & % off, see the saving
Open →Heart rate
Class 10 Biology
Set heart rate & time, count the beats
Open →Resistors in parallel
Class 10 Physics
Wire two resistors side by side
Open →Electric charge
Class 10 Physics
Flow current over time, collect charge
Open →Electrical energy and units
Class 10 Physics
Run appliances, add up the units
Open →Kelvin temperature scale
Class 9 Chemistry
Slide Celsius, read the Kelvin
Open →Moles from number of particles
Class 9 Chemistry
Divide particles by Avogadro's number
Open →Ten percent law
Class 10 Biology
See 10% of energy reach the next level
Open →Area of a square
Class 8 Maths
Grow the side, square the area
Open →Volume of a cube
Class 8 Maths
Grow the edge, cube the volume
Open →Circumference of a circle
Class 8 Maths
Grow the radius, roll out the rim
Open →Surface area of a cube
Class 9 Maths
Grow the edge, cover six faces
Open →Potential difference
Class 10 Physics
Share work across charge, get volts
Open →Resistance from Ohm's law
Class 10 Physics
Divide voltage by current, get resistance
Open →Echo and SONAR
Class 9 Physics
Time the echo, find the distance
Open →Mass from moles
Class 9 Chemistry
Multiply moles by molar mass
Open →Breathing rate
Class 10 Biology
Set breathing rate & time
Open →Volume of a cylinder
Class 10 Maths
Set radius & height, fill the can
Open →Compound interest
Class 8 Maths
Compound money over years
Open →Profit and loss percentage
Class 8 Maths
Set cost & selling price, see profit %
Open →Perimeter of a rectangle
Class 8 Maths
Set length & breadth, walk the border
Open →Surface area of a sphere
Class 10 Maths
Grow the radius, wrap the ball
Open →Time period
Class 9 Physics
Raise the frequency, shrink the period
Open →Relative velocity
Class 9 Physics
Two objects approach - add their speeds
Open →Average velocity
Class 9 Physics
Average the start and end speeds
Open →Equations of motion (v = u + at)
Class 9 Physics
Accelerate from u for a time t
Open →Kelvin to Celsius
Class 9 Chemistry
Slide Kelvin, read the Celsius
Open →Population growth rate
Class 10 Biology
Balance births vs deaths per population
Open →Perimeter of a square
Class 8 Maths
Grow the side, walk four edges
Open →Perimeter of a triangle
Class 8 Maths
Add the three sides
Open →Area of a parallelogram
Class 8 Maths
Set base & height, slide the shape
Open →Area of a rhombus
Class 8 Maths
Set the two diagonals
Open →Equations of motion (distance)
Class 9 Physics
Start, accelerate, cover ground
Open →Joule's law of heating
Class 10 Physics
Raise current, resistance or time
Open →Electric power (P = VI)
Class 10 Physics
Multiply voltage by current
Open →Average atomic mass of isotopes
Class 9 Chemistry
Mix two isotopes by abundance
Open →Seed germination percentage
Class 9 Biology
Count sprouted seeds out of the total
Open →Volume of a cone
Class 9 Maths
Set radius & height, fill the cone
Open →Surface area of a cylinder
Class 9 Maths
Wrap the side and both ends
Open →Surface area of a cuboid
Class 9 Maths
Cover all six rectangular faces
Open →nth term of an AP
Class 10 Maths
Step from the first term by d
Open →Sum of an AP
Class 10 Maths
Add up the first n terms
Open →Focal length of a mirror
Class 10 Physics
Halve the radius to find the focus
Open →Speed of light in a medium
Class 10 Physics
Raise the index, slow the light
Open →Percentage purity
Class 9 Chemistry
Weigh the pure part of a sample
Open →Slope of a line
Class 10 Maths
Rise over run gives the steepness
Open →Percentage change
Class 8 Maths
Compare a new value to the old
Open →Volume of a hemisphere
Class 9 Maths
Grow the radius of half a ball
Open →Area of a sector
Class 10 Maths
Cut a slice of angle from a circle
Open →Length of an arc
Class 10 Maths
Measure the curved edge of a slice
Open →Slant height of a cone
Class 9 Maths
Combine radius & height for the slant
Open →Unit conversion (km/h to m/s)
Class 9 Physics
Convert km/h into m/s
Open →Equations of motion (v^2 = u^2 + 2as)
Class 9 Physics
Accelerate over a distance, find v
Open →Impulse
Class 9 Physics
Hit harder or longer, change momentum
Open →Wavelength
Class 9 Physics
Speed over frequency gives wavelength
Open →Oscillations
Class 9 Physics
Vibrate at a frequency for a time
Open →Cost of electricity
Class 10 Physics
Units times rate gives the bill
Open →Number of neutrons
Class 9 Chemistry
Take protons away from the mass number
Open →Curved surface area of a cone
Class 9 Maths
Wrap the slanted side of a cone
Open →Total surface area of a cone
Class 9 Maths
Add the base circle to the cone's side
Open →Curved surface area of a hemisphere
Class 9 Maths
Cover the dome of a hemisphere
Open →Diagonal of a square
Class 9 Maths
Cross a square corner to corner
Open →Unit conversion (m/s to km/h)
Class 9 Physics
Convert m/s into km/h
Open →Distance from velocities
Class 9 Physics
From two speeds, find the distance
Open →Diagonal of a rectangle
Class 9 Maths
Cross a rectangle corner to corner
Open →Diagonal of a cuboid
Class 9 Maths
The longest rod that fits in a box
Open →Area by Heron's formula
Class 9 Maths
Area from just the three sides
Open →Interior angle sum of a polygon
Class 8 Maths
Add up a polygon's inside angles
Open →Exterior angle of a regular polygon
Class 8 Maths
Share 360 among a polygon's corners
Open →Number of diagonals of a polygon
Class 8 Maths
Count the diagonals of a polygon
Open →Discriminant
Class 10 Maths
Test how many roots a quadratic has
Open →Sum of roots
Class 10 Maths
Sum of a quadratic's roots
Open →Punnett square (monohybrid cross)
Class 10 Biology
Cross two parents, predict the offspring
Open →Balancing chemical equations
Class 10 Chemistry
Slide coefficients until atoms balance
Open →Writing chemical formulae (valency)
Class 9 Chemistry
Criss-cross valencies into a formula
Open →Current from power
Class 10 Physics
Divide power by voltage for current
Open →Power (P = V^2 / R)
Class 10 Physics
Voltage squared over resistance
Open →Sine ratio
Class 10 Maths
Opposite over hypotenuse
Open →Cosine ratio
Class 10 Maths
Adjacent over hypotenuse
Open →Tangent ratio
Class 10 Maths
Opposite over adjacent
Open →Area of an equilateral triangle
Class 9 Maths
Area of an equilateral triangle
Open →Curved surface area of a cylinder
Class 9 Maths
Wrap only the curved side
Open →Loss percentage
Class 8 Maths
Sell below cost, find the loss %
Open →Amount with simple interest
Class 8 Maths
Principal plus its simple interest
Open →Distance formula
Class 10 Maths
Straight distance between two points
Open →States of matter
Class 9 Chemistry
Heat particles solid → liquid → gas
Open →Parts of a plant cell
Class 8 Biology
Tap a cell part to see its job
Open →Diagonal of a cube
Class 9 Maths
Longest diagonal through a cube
Open →Total surface area of a hemisphere
Class 9 Maths
Dome plus its flat circle
Open →Sum of first n natural numbers
Class 10 Maths
Add 1 + 2 + ... + n instantly
Open →Range of data
Class 9 Maths
Spread from smallest to largest
Open →Class mark
Class 9 Maths
Midpoint of a class interval
Open →Selling price from profit percent
Class 8 Maths
Mark up cost by a profit %
Open →Perimeter of a sector
Class 10 Maths
Two radii plus the curved arc
Open →Circumference from diameter
Class 8 Maths
Circumference straight from diameter
Open →Power (P = F x v)
Class 9 Physics
Force times velocity gives power
Open →Percentage of a number
Class 8 Maths
Find a percentage of a number
Open →Series and parallel circuits
Class 10 Physics
Break a bulb in series vs parallel
Open →Symbols of elements
Class 9 Chemistry
Match each element to its symbol
Open →Free fall (velocity)
Class 9 Physics
Drop from a height, hit this speed
Open →Free fall (time)
Class 9 Physics
How long a drop takes
Open →Free fall (distance)
Class 9 Physics
Distance fallen in a given time
Open →Complement of an event
Class 10 Maths
Chance an event does NOT happen
Open →Product of roots
Class 10 Maths
Product of a quadratic's roots
Open →Exterior angle theorem
Class 9 Maths
Exterior angle = sum of remote interiors
Open →Complementary angles
Class 10 Maths
What adds to 90 degrees
Open →Supplementary angles
Class 9 Maths
What adds to 180 degrees
Open →Perimeter of a semicircle
Class 10 Maths
Curved half plus the diameter
Open →Area of a semicircle
Class 10 Maths
Half the area of a circle
Open →Turning effect (moments)
Class 9 Physics
Balance the see-saw with moments
Open →Reflex arc
Class 10 Biology
Step through a reflex, stimulus to action
Open →Buoyant force (upthrust)
Class 9 Physics
Displace liquid, feel the upthrust
Open →Relative density
Class 9 Physics
Compare a density to water's
Open →Power in lifting a load
Class 9 Physics
Lift a load, faster needs more power
Open →Cosecant ratio
Class 10 Maths
Hypotenuse over opposite
Open →Secant ratio
Class 10 Maths
Hypotenuse over adjacent
Open →Cotangent ratio
Class 10 Maths
Adjacent over opposite
Open →Height from angle of elevation
Class 10 Maths
Height from an angle of elevation
Open →Area of a quadrant
Class 10 Maths
A quarter of a circle's area
Open →Interior angle of a regular polygon
Class 8 Maths
One inside angle of a regular polygon
Open →Sum of first n odd numbers
Class 10 Maths
Add the first n odd numbers
Open →Sum of first n even numbers
Class 10 Maths
Add the first n even numbers
Open →Quadratic formula (a root)
Class 10 Maths
Larger root of a quadratic
Open →LCM from HCF
Class 10 Maths
LCM from the product and HCF
Open →Depreciation
Class 8 Maths
Value drops by a % each year
Open →Cost price from selling price
Class 8 Maths
Work back to the cost price
Open →Downstream speed
Class 8 Maths
Row with the current
Open →Upstream speed
Class 8 Maths
Row against the current
Open →Average speed for a round trip
Class 8 Maths
Average speed there and back
Open →Sales tax / GST
Class 8 Maths
Tax added on a price
Open →Area of a ring (annulus)
Class 10 Maths
Area of a ring between two circles
Open →Edge of a cube from volume
Class 9 Maths
Edge back from the volume
Open →Radius from area
Class 10 Maths
Radius back from a circle's area
Open →Side from area of a square
Class 8 Maths
Side back from a square's area
Open →Height of a triangle from area
Class 9 Maths
Height back from area and base
Open →Rate from simple interest
Class 8 Maths
Rate back from the interest
Open →Time from simple interest
Class 8 Maths
Time back from the interest
Open →Principal from simple interest
Class 8 Maths
Principal back from the interest
Open →Mean proportional
Class 10 Maths
Geometric mean of two numbers
Open →Fourth proportional
Class 8 Maths
Complete the proportion a : b = c : ?
Open →Marked price from selling price
Class 8 Maths
Marked price back from the sale price
Open →Chambers of the human heart
Class 10 Biology
Tap a heart chamber to see its job
Open →Equation of a line (y = mx + c)
Class 9 Maths
Read y off a straight line
Open →Average term of an AP
Class 10 Maths
Average of first and last term
Open →Number of terms in an AP
Class 10 Maths
How many terms in an AP
Open →Midpoint of two points
Class 10 Maths
x-coordinate of a midpoint
Open →Empirical mode
Class 10 Maths
Estimate the mode from mean & median
Open →Length of a shadow
Class 10 Maths
Shadow from height and sun angle
Open →Train crossing a pole
Class 8 Maths
Speed to cross a pole
Open →Time and work
Class 8 Maths
More workers, fewer days
Open →Dividing in a ratio
Class 8 Maths
Split a total in a ratio
Open →Unitary method
Class 8 Maths
Cost of a single item
Open →Your progress — Foundation - Class 8 to 10
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Showing 18 questions for Class 9. Tap a card to reveal the answer.
MathematicsSurface Areas and VolumesmediumFind 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.
V = 4/3 π r³ ≈ 1437.3 cm³
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 Areas and VolumeseasyFind 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.
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.
MathematicsSurface Areas and VolumesmediumFind 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.
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 Areas and VolumesmediumFind 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.
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 Areas and VolumesmediumFind 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.
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).
MathematicsSurface Areas and VolumesmediumFind 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.
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.
MathematicsSurface Areas and VolumesmediumA 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.
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.
MathematicsSurface Areas and VolumesmediumFind 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.
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).
MathematicsSurface Areas and VolumesmediumFind 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.
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).
MathematicsSurface Areas and VolumeseasyFind 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.
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.
MathematicsSurface Areas and VolumesmediumFind 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.
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).
MathematicsSurface Areas and VolumeseasyFind 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.
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.
MathematicsSurface Areas and VolumesmediumFind 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.
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.
MathematicsSurface Areas and VolumesmediumFind 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.
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).
MathematicsSurface Areas and VolumesmediumA 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.
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.
MathematicsSurface Areas and VolumesmediumFind 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 Areas and VolumesmediumFind 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.
MathematicsSurface Areas and VolumesmediumFind 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).
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