Find the area of the base, then decide whether to take all of the height or a third of it.
Find the volume
A tent is a square pyramid with a base 6 m by 6 m and a height of 4 m. What is its volume?
The base area is 6 × 6 = 36 m², and a pyramid is a third of the box around it: (13) × 36 × 4 = 48 m³. C forgets the third and gives the whole box, 144 m³. D multiplies all three numbers and the base twice over. A divides by 6 instead of 3.
Volume = base area × height (and a third of that if it comes to a point)
| Shape | Base area | Volume |
|---|---|---|
| rectangular prism | l × w | lwh |
| cylinder | πr² | πr²h |
| rectangular pyramid | l × w | lwh3 |
| square pyramid | s² | s²h3 |
There is really one formula here, not four. Work out the area of the flat face it stands on, multiply by how tall it is, and divide by 3 only if it narrows to a point.
Stack identical circles and you have a can.
A tin can labelled "14 cm across" has radius 7, not 14. Squaring the wrong one makes the answer four times too big.
It is not an approximation, and it is not a half.
Three pyramids with the same base and the same height fill their box exactly. So one pyramid is one third of it — hence the 3 underneath.
base = 6 × 6 = 36
the tent from the opener
box = 36 × 4 = 144
if it were a prism
V = 144 ÷ 3 = 48
but it comes to a point
Height means straight up from the base to the tip.
The slant is always the longer of the two, so using it inflates the volume. It has its own job — surface area — but never volume.
When the unknown is squared in the formula, solve for the square first, then take its root.
A square pyramid has a volume of 147 cm³ and a height of 9 cm. How long is each side of its base?
147 = 9s²3
V = s²h3
147 = 3s², so s² = 49
9 ÷ 3 = 3, then divide by 3
s = √49 = 7 cm
check: 7 × 7 × 9 ÷ 3 = 147 ✓
49 is the area of the base, not its side. A side of 49 cm would give a volume of 49 × 49 × 9 ÷ 3 = 7,203 cm³. The last move is always the square root.
A cylindrical water tank has a radius of 0.5 m and a height of 2 m. How many litres does it hold? Use π ≈ 3.14 and 1 m³ = 1,000 L.
3.14 × 0.5² × 2 = 3.14 × 0.25 × 2 = 1.57 m³
V = πr²h, in cubic metres
1.57 × 1,000 = 1,570 L
then convert
Scaling: a cylinder's radius is tripled and its height kept the same. What happens to its volume?
π(3r)²h = π × 9r² × h
the 3 is squared along with r
= 9 × πr²h
9 times the volume, not 3
Tripling the height instead would only triple the volume, because h is not squared.
Still base area × height; these change which height or which radius goes in.
A can of radius 4 cm and height 15 cm is filled with water to a depth of 9 cm. Find the volume of water, using π ≈ 3.14.
3.14 × 4² × 9
the water is a cylinder 9 cm tall
= 3.14 × 144 = 452.16 cm³
16 × 9 = 144
Use the depth of the water, not the height of the can; the empty part holds nothing.
A cylinder is 8 cm across and 5 cm tall. Give its volume in terms of π.
r = 8 ÷ 2 = 4 cm
"across" is the diameter
V = π × 4² × 5 = 80π cm³
keep π as a letter
An exact answer in π skips the rounding, so the halving step is the only place left to slip.
A pyramid fits exactly inside a cube of edge 6 cm: its base is the cube's floor and its tip touches the lid. Find its volume.
base 6 × 6 = 36, height 6
the height is the cube's edge
V = 36 × 6 ÷ 3 = 72 cm³
a third of the cube's 216 cm³
The leftover space in the cube is 216 − 72 = 144 cm³, two thirds of it.
Put the numbers you have into the formula and solve for the letter left over.
V = 300 cm³, base 10 by 5
a rectangular pyramid; find h
300 = 50h3
base area is 50
900 = 50h
multiply both sides by 3
h = 18 cm
divide by 50
Check: 50 × 18 ÷ 3 = 300 ✓
Volume items give a cylinder or a pyramid with a diameter or a slant height printed as bait, ask for the answer in terms of π, or give the volume and ask for a missing height or side. The fastest route is base area first, then × height, then ÷ 3 if it comes to a point; the 10-second check is to compare a pyramid with its box — it must be exactly a third.
Find the height
A cylindrical drum has a radius of 5 cm and a volume of 200π cm³. How tall is it?
V = πr²h, so 200π = π(5)²h = 25πh. Divide both sides by 25π: h = 8 cm. Check: 25π × 8 = 200π ✓ D divides by 5 instead of 25 — it forgets to square the radius. A divides by 50, treating πr² as 2πr.
Hint: the π appears on both sides, so it cancels — you never need a decimal value for it here.
