Three formulas, one idea: how much of its cylinder does the shape actually fill?
Compare the two
A cone and a sphere both have a radius of 6 cm, and the cone's height is also 6 cm. Which holds more, and by how much?
Cone: πr²h3 = π(36)63 = 72π cm³. Sphere: 4πr³3 = 4π(216)3 = 288π cm³. The sphere holds 288π − 72π = 216π cm³ more — four times as much. B is the cone's own volume, not the difference. A sphere is far bigger than most people expect, because the radius is cubed.
A cone is a third of its cylinder. A sphere is two thirds of its cylinder.
| Shape | Volume | Why |
|---|---|---|
| cylinder | πr²h | base area × height |
| cone | πr²h3 | a third of its cylinder |
| pyramid | base area × h3 | a third of its prism |
| sphere | 4πr³3 | two thirds of a cylinder of height 2r |
Check the sphere claim: a cylinder around a sphere has height 2r, so its volume is πr²(2r) = 2πr³. Two thirds of that is 4πr³3 ✓
r = 6 cm, h = 8 cm
an ice-cream cone
base = π(6)² = 36π
square the radius first
cylinder = 36π × 8 = 288π
if it had straight sides
cone = 288π ÷ 3 = 96π
≈ 301 cm³
A cone with r = 6 and h = 8 has a slant height of 10, and problems often print it. Volume uses the perpendicular height, 8. The slant is for surface area only.
And that radius is cubed, so small changes matter enormously.
r = 4 cm
a small ball
r³ = 64
cube it, do not multiply by 3
V = 4π(64)3 = 256π3
≈ 268 cm³
Doubling the radius makes a sphere eight times bigger, because 2³ = 8. A ball twice as wide holds eight times the air.
Undo the formula one operation at a time, cube root last.
V = 288π cm³, a sphere
find r
288π = 4πr³3
substitute
864π = 4πr³
× 3
r³ = 216
÷ 4π — the π cancels
r = 6 cm
cube root
Which matches the opening problem — that sphere had radius 6 and volume 288π ✓
A hemisphere is half a sphere, so halve the sphere formula. Split any awkward solid into shapes you already have formulas for, work each out, then add.
Pyramid: base area × height ÷ 3, whatever shape the base is.
A pyramid has a rectangular base 6 cm by 5 cm and a height of 9 cm. Find its volume.
base area = 6 × 5 = 30 cm²
a rectangle
prism = 30 × 9 = 270 cm³
the box around it
pyramid = 270 ÷ 3 = 90 cm³
a third of its prism
A cone has radius 5 cm and height 6 cm. Find its volume, taking π as 3.14.
πr²h3 = 3.14 × 25 × 6 ÷ 3
substitute
= 3.14 × 50 = 157 cm³
25 × 6 ÷ 3 = 50 first
Do the whole-number part first and multiply by 3.14 last — fewer decimals to carry. In terms of π the answer is 50π cm³.
The same formulas, with the given and the asked swapped round.
A sphere has diameter 18 cm. Find its volume in terms of π.
r = 18 ÷ 2 = 9
halve the diameter first
r³ = 729
9 × 9 × 9
V = 4π(729)3 = 972π cm³
729 ÷ 3 = 243, × 4
Using 18 as the radius gives eight times too much. Check which length the question gives.
A square pyramid has base side 6 cm and volume 120 cm³. How tall is it?
120 = 36 × h3
base area 6 × 6 = 36
120 = 12h
36 ÷ 3 = 12
h = 10 cm
Check: 36 × 10 ÷ 3 = 120 ✓.
A cone-shaped container has radius 10 cm and height 21 cm. How many litres does it hold? Use π = 227.
V = 227 × 100 × 21 ÷ 3
πr²h3
= 22 × 100 × 3 ÷ 3 = 2,200 cm³
21 ÷ 7 = 3
= 2.2 L
1 L = 1,000 cm³
Choose 227 when a measurement is a multiple of 7; it cancels neatly.
Items give a radius and a height (often with a slant height printed as a decoy) and ask for the volume, ask how the volume changes when the radius doubles, or give the volume and ask for a missing length. The fastest route is to pick the formula, square or cube the radius, and keep π as a symbol unless told to use 3.14. For a 10-second check, a cone must come out a third of its cylinder, and π ≈ 3 gives a quick estimate.
Find the height
A cone has a radius of 4 cm and a volume of 48π cm³. How tall is it?
V = πr²h3, so 48π = π(16)h3 = 16πh3. Multiply by 3: 144π = 16πh, and dividing by 16π gives h = 9 cm. Check: π(16)93 = 144π3 = 48π ✓ A forgets to multiply by 3, treating the cone as a cylinder.
Hint: π cancels from both sides, so it never needs a decimal value. Clear the division by 3 before dividing by anything else.
