Solids & scaling
Volumes, and why doubling a size more than doubles its weight.
Pen and paper is fine · no calculator needed why?
Opens at level 26.
the lesson
Read the lesson
The idea
Volume is the space inside a solid. A cylinder's volume is its base area times its height, πr²h. A cone with the same base and height holds exactly a third of that. A sphere holds (4/3)πr³. As with circles, answers are written as a number times π.
Scaling is where intuition fails. Make every length k times longer and areas grow k × k times, while volumes, and so weights, grow k × k × k times. A statue twice as tall as another of the same shape and material weighs eight times as much.
Techniques
Base area times height
- Base: square the radius. The π stays outside.
- Cylinder: multiply by the height.
- Cone: multiply by the height, then divide by 3.
worked example
A cone has base radius 4 and height 9. Its volume is kπ. What is k?
- Base: 4 × 4 = 16.
- Times the height: 16 × 9 = 144.
- A cone is a third: 144 ÷ 3 = 48.
Answer: 48
Sphere: cube, then four thirds
- Given the diameter? Halve it to get the radius.
- Cube the radius: r × r × r.
- Multiply by 4 and divide by 3. A fraction answer is fine.
worked example
A sphere has radius 6. Its volume is kπ. What is k?
- 6 × 6 × 6 = 216.
- 216 × 4 = 864, and 864 ÷ 3 = 288.
Answer: 288
Square for area, cube for volume
- Find the length factor: how many times longer the big one is.
- Areas scale by the factor squared.
- Volumes, and weights of the same material, scale by the factor cubed.
- Going from big to small, divide instead of multiplying.
worked example
A small tank holds 3 litres. A larger tank has the same shape, with every length 4 times as long. How many litres does the larger tank hold?
- Volume scales by 4 × 4 × 4 = 64.
- 3 × 64 = 192 litres.
Answer: 192
Tips by skill
- TipCylinders & cones: Square the radius and multiply by the height. For a cone, divide that by 3.
- TipArea scale factor: Areas scale by the square of the length factor: multiply going up, divide going down.
- TipSpheres: Halve a diameter first. Cube the radius, multiply by 4, and divide by 3.
- TipVolume scale factor: Volume and weight scale by the cube of the length factor: multiply going up, divide going down.
Watch out for
- Working out the side area 2πrh instead of the volume πr²h.
- Forgetting the one third for a cone, or the four thirds for a sphere.
- Scaling an area or a weight by the plain length factor. A model at 1/10 the size weighs 1/1,000 as much, not 1/10.
- Using the diameter as the radius.
skills · practice stats
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Cylinders & cones not tried yet
worked example
A cylinder has radius 1 and height 9. Its volume is kπ. What is k?
Answer: 9
- V = πr²h = π × 1 × 9 = 9π, so k = 9.
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Area scale factor not tried yet
worked example
Two shapes are similar: every length of the big one is 3 times the matching length of the small one. The small one has an area of 10 cm². What is the big one's area, in cm²?
Answer: 90
- Areas scale by the square of the length factor: 3² = 9.
- 10 × 9 = 90 cm².
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Spheres not tried yet
worked example
A sphere has diameter 4. Its volume is kπ. What is k? Give k as a whole number or a fraction.
Answer: 32/3
- The radius is half the diameter: 2.
- V = (4/3)πr³ = (4/3) × 8 × π = (32/3)π, so k = 32/3.
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Volume scale factor not tried yet
worked example
A bell weighs 5,632 kg. A model of it is built at 1/8 of its size in every length, from the same material. How many kilograms does the model weigh?
Answer: 11
- Weight scales like volume, by the cube of the length factor: 8³ = 512.
- 5,632 ÷ 512 = 11 kg.
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