Volume of Prisms and Cylinders

How much fits inside?

How much water fills the tank on a roof? How much juice hides in a can? How much concrete goes into a steel beam? All three questions are secretly the same question — and they all have the same delightfully simple answer.

The tank, the can and the beam are each a prism: a solid whose shape stays exactly the same the whole way through, like toothpaste squeezed out in one long strip. Slice any of them across and you always meet the identical flat shape — a circle for the can, a rectangle for the beam. That one repeated shape is the secret. Find its area, multiply by how long the solid runs, and you have the volume. That's it.

V = (\text{area of the cross-section}) \times \text{length}

A prism: the same shape all the way along

A prism is a solid with the same cross-section all along its length — slice it anywhere across and you always get the same flat shape. Because every slice is identical, the volume is simply that one cross-section's area, repeated all the way along:

V = (\text{area of cross-section}) \times \text{length}

A cuboid (a box) is a prism whose cross-section is a rectangle. Its cross-section has area l \times w, so stretching it along a height h gives

V = l \times w \times h

Think of it as stacking flat sheets. One rectangle has area l \times w but no thickness. Stack h of those sheets on top of each other and you build a solid box — and its volume is just that area, taken h deep. The very same trick works for any cross-section, not just rectangles.

A cylinder is a prism too

A cylinder is a prism whose cross-section is a circle. The circle of radius r has area \pi r^2, so carrying it along a height h gives

V = \pi r^2 \times h = \pi r^2 h

A soup can, a drainpipe, a roll of coins, a tree trunk — every one is (near enough) a cylinder, and every one obeys this single rule.

For any solid with a constant cross-section:

Prism: V = (\text{cross-section area}) \times \text{length}.
Cuboid (rectangular cross-section): V = l \times w \times h.
Cylinder (circular cross-section): V = \pi r^2 h.

Worked example 1 — a triangular prism

A chocolate bar comes in a long triangular-prism box. The triangular end has base 6\text{ cm} and height 4\text{ cm}, and the box is 20\text{ cm} long. What volume of chocolate does it hold?

Step 1 — the cross-section area. The end is a triangle:

A = \tfrac{1}{2} \times \text{base} \times \text{height} = \tfrac{1}{2} \times 6 \times 4 = 12\text{ cm}^2

Step 2 — multiply by the length (the direction perpendicular to that triangular face):

V = 12 \times 20 = 240\text{ cm}^3

So the box holds 240\text{ cm}^3 of chocolate. Notice we found the cross-section area once, then just dragged it along the length. Same recipe every time.

Worked example 2 — a can (exact and rounded)

A tin of beans is a cylinder with radius 5\text{ cm} and height 11\text{ cm}. What is its volume?

V = \pi r^2 h = \pi \times 5^2 \times 11 = \pi \times 25 \times 11 = 275\pi\text{ cm}^3

The exact answer is 275\pi\text{ cm}^3 — leaving the \pi in keeps it perfectly precise. To get a decimal answer, use \pi \approx 3.142:

V \approx 275 \times 3.142 \approx 864\text{ cm}^3

A neat habit: work out the number in front of \pi first (25 \times 11 = 275), and multiply by \pi only at the very end. It keeps the arithmetic clean and lets you give either answer.

Worked example 3 — how many litres?

A cylindrical water butt has radius 30\text{ cm} and height 100\text{ cm}. Roughly how many litres does it hold?

Step 1 — the volume in cm³:

V = \pi r^2 h = \pi \times 30^2 \times 100 = \pi \times 900 \times 100 = 90000\pi \approx 282740\text{ cm}^3

Step 2 — turn cm³ into litres. One litre is exactly 1000\text{ cm}^3 (a cube 10\text{ cm} on each side), so divide by 1000:

282740 \div 1000 \approx 283\text{ litres}

That's a big barrel! Getting the units right at the end is just as important as the formula — a volume in \text{cm}^3 is a thousand times bigger a number than the same volume in litres.

1\text{ litre} = 1000\text{ cm}^3 (a 10 cm cube).
1\text{ millilitre} = 1\text{ cm}^3 — so a can of about 330\text{ cm}^3 is a 330 ml can. That's why cans say "330 ml"!
1\text{ m}^3 = 1000000\text{ cm}^3 = 1000\text{ litres} — a cubic metre of water is a whole tonne.

Two prisms to picture

Step through the sketch: first a cuboid (rectangular cross-section), then a cylinder (circular cross-section). Each is a flat drawing of a 3D solid.

Three slips catch almost everybody with prisms and cylinders:

Use the cross-section that stays constant. It's the face that repeats all the way through — and you multiply by the length perpendicular to it. For a can lying on its side, the constant shape is still the circular end, not the rectangle you see; multiply the circle's area by the can's length. Pairing the wrong face with the wrong length gives nonsense.
r is the radius, not the diameter. If a pipe is 10\text{ cm} across, that's the diameter — the radius is 5\text{ cm}. Putting 10 into \pi r^2 makes the answer four times too big, because the radius is squared. This is the single most common mistake.
Keep your units straight. \text{cm}^3, litres and \text{m}^3 are all volumes but very different-sized numbers. Decide which one the question wants before you start, and convert only at the end.

They're all extruded — pushed through a shaped hole. Squeeze soft pasta dough through a star-shaped hole and out comes an endless star-cross-section noodle; cut it to length and every piece is a prism. Aluminium window frames, railway rails, the bars of a LEGO-brick sprue, even the humble drinking straw — all made by forcing material through a die with the desired cross-section.

This is exactly why the volume rule is so simple. Extrusion is "take a flat shape and drag it along a length". So the amount of material is just the area of that shape times how far you dragged it. Factories don't think about volume formulas — but the metal obeys V = A \times \text{length} anyway.

Take a straight stack of coins and gently push it sideways so it leans, like the Tower of Pisa. Not a single coin has changed, so the volume is exactly the same — even though the leaning stack looks quite different. This is Cavalieri's principle: if two solids have matching cross-sections at every height, they have equal volume. It's why a slanted (oblique) prism holds the same as an upright one, as long as you use the perpendicular height.

Cavalieri worked this out in the 1600s by imagining a solid as infinitely many wafer-thin slices. That "add up all the slices" idea grows up into integration, which lets you find the volume of shapes far weirder than prisms.