KS3 · Maths

Volume of a cylinder

A cylinder is a pile of identical coins: find the area of one coin, multiply by how tall the pile is, and you've found its volume.

Build the formula

Stack the circles

Height: 4/10. Height 4 cm. Area of each slice 9π cm² (≈ 28.3 cm²). Volume (exact) 36π cm³. Volume (1 d.p.) 113.1 cm³

Height4 cmArea of each slice9π cm² (≈ 28.3 cm²)Volume (exact)36π cm³Volume (1 d.p.)113.1 cm³

This cylinder is lying on its side so you can see it slice by slice. Its radius is 3 cm, so every circular slice has area π × 3² = 9π cm², which is about 28.3 cm². Drag the end to make it longer. The slice never changes, and every extra 1 cm of height adds one more slice's worth of volume: another 9π cm³.

Sort the solids

Which solids stack like a cylinder?

Imagine slicing each solid straight across, parallel to its base. Tick every set it belongs to. Some go in both, some in one, and some in neither.

A Every slice is identical (same shape, same size)
B The slice is a polygon (straight sides only)

Cylinder (a tin can)
Cuboid
Cube
Triangular prism
Hexagonal prism (like a pencil)
Square-based pyramid
Cone
Sphere (a ball)

Worked example: a vase

Problem

A cylindrical vase has diameter 8 cm and height 0.15 m. How much space is inside it? Treat it as a cylinder and ignore the thickness of the glass. Give the volume exactly, in terms of π, and then to 1 decimal place.

What would you do?

Four ways to read πr²h

A cylinder has diameter 10 cm and height 4 cm. Four learners start working out its volume in different ways.

Which method is closest to what you would do?

Put r = 10 straight in: π × 10² × 4 = 400π cm³.Use r = 5, work out π × 5 first, then square it: (π × 5)² × 4.Use r = 5, multiply π × 5 × 4 first, then square the whole lot: (π × 5 × 4)².Halve the diameter to get r = 5, square it to get 25, then multiply by π and by 4: 100π cm³.

How sure are you?

I'm sureI think soI'm guessing

Put r = 10 straight in: π × 10² × 4 = 400π cm³.

This is a common misconception

10 cm is the diameter, all the way across. The formula needs the radius, from the centre to the edge, which is half of that: 5 cm. Using the diameter makes the circle's area 4 times too big (10² = 100 instead of 5² = 25), so the volume comes out 4 times too big.

Use r = 5, work out π × 5 first, then square it: (π × 5)² × 4.

This is a common misconception

That squares π as well as the radius. Powers come before multiplying, so in πr²h only r is squared. Squaring π too gives 100π², about 987 cm³, which is more than three times the real answer.

Use r = 5, multiply π × 5 × 4 first, then square the whole lot: (π × 5 × 4)².

This is a common misconception

The ² is attached to r only, not to the whole product. Squaring everything squares π and the height too, giving 400π², nearly 3950 cm³. That's more than twelve times too big.

Halve the diameter to get r = 5, square it to get 25, then multiply by π and by 4: 100π cm³.

This matches the evidence

Exactly right. Halve first, square only the radius, then multiply by π and the height. 100π cm³ is exact, or about 314.2 cm³ if you need a decimal.

Halve a diameter first. Then square only the radius, and multiply by π and the height.

Now run it backwards

Find the diameter from the volume

A cylinder has volume 392π cm³ and height 8 cm. What is its diameter?

Put what you know into V = πr²h: 392π = π × r² × 8This time r is the unknown.
missing step

Which line is step 2?

Divide both sides by π: 392 = 8 × r²Subtract π from both sides: 392π − π = 8 × r²Divide both sides by π: 392 = 8 × r

Hollow cylinders: two routes, one answer

Route 1: ring × lengthvsRoute 2: big cylinder − hole

A metal pipe is a hollow cylinder: outer radius 5 cm, inner radius 4 cm, length 12 cm. How much metal is in it?

Focus

The idea

Route 1: ring × length

Slice the pipe. Every slice is the same ring, so find the ring's area and stack it 12 cm long.

Route 2: big cylinder − hole

Pretend the pipe is solid, find that volume, then take away the cylinder of air in the middle.

The insight

Both routes are cross-section × length thinking. Route 1 stacks rings; Route 2 stacks circles twice and subtracts.

First calculation

Route 1: ring × length

Ring area = π × 5² − π × 4² = 25π − 16π = 9π cm²

Route 2: big cylinder − hole

Solid cylinder = π × 5² × 12 = 300π cm³

Second calculation

Route 1: ring × length

Volume = 9π × 12 = 108π cm³

Route 2: big cylinder − hole

Hole = π × 4² × 12 = 192π cm³, so 300π − 192π = 108π cm³

What you need

Route 1: ring × length

Both radii and the length

Route 2: big cylinder − hole

Both radii and the length

Easy slip

Route 1: ring × length

Subtracting the radii before squaring

Route 2: big cylinder − hole

Stopping at 300π and forgetting to subtract the hole

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find the area of one circle, stack it up to the height, and you've found the volume.

What you need to know

Volume is the amount of space inside a 3D solid. It's measured in cubic units such as cm³, mm³ or m³.
A cylinder has the same circular cross-section all the way through, like a stack of identical coins.
Volume of a cylinder = area of the circle × height, so V = πr²h.
In the formula, height, length and depth all mean the same thing: how far the circle is stacked.

The big picture

A cylinder is a stack of identical circles, so its volume is the area of its circular cross-section times its height: V = πr²h, measured in cubic units. Halve a diameter first, square only the radius, and you can use the formula forwards, backwards and for hollow cylinders.

Key points

1If you're given a diameter, halve it to get the radius before you do anything else.

2Put every length in the same unit before you calculate.

3Square only the radius, then multiply by π and by the height.

4Leaving π in the answer (like 240π cm³) is exact. Round only when asked, to the accuracy asked.

5Working backwards: divide by the cross-section area to find the height. To find the radius, divide by π and the height, then square-root.

6Hollow cylinder: ring area × height, or big cylinder − inner cylinder. Both give the same answer.

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Worked example

Problem

A cylindrical tin holds 720π cm³ and has radius 6 cm. How tall is it?

⚠ Watch out

Using the diameter as r, or squaring π along with the radius. Halve a diameter first, then square only the radius before multiplying by π and the height.

🧠

Memory hook

One coin, then the pile: find the area of one circle (πr²), then stack it h high.

✓

Check yourself

In one sentence, explain why a cylinder's volume is πr²h, using the words 'cross-section' and 'height'. Then find the volume when the diameter is 6 cm and the height is 10 cm. (Answer: 90π cm³.)

Flashcards

(14)

What is volume?

The amount of space inside a 3D solid.

Why is volume measured in cubic units, like cm³?

It comes from multiplying three lengths at right angles: cm × cm × cm.

What does it mean that a cylinder has a uniform cross-section?

Every slice across it, parallel to its ends, is the same circle, like a stack of identical coins.

Why isn't a cylinder strictly a prism?

A prism's cross-section is a polygon (straight sides only), and a circle isn't a polygon.

How do you find the volume of any solid that has the same slice all the way through?

Area of the cross-section × height (or length).

What is the area of a cylinder's cross-section?

πr², the area of its circular end.

In πr²h, which part is squared?

Only r. Square the radius, then multiply by π and by h.

You're given a cylinder's diameter. What's your first step?

Halve it to get the radius.

One length is in cm and another is in m. What must you do before calculating?

Convert them so every length is in the same unit.

Why might you leave a volume 'in terms of π'?

It's exact, with no rounding error.

How do you find a cylinder's height from its volume and radius?

Divide the volume by the cross-section area, πr².

How do you find a cylinder's radius from its volume and height?

Divide by π and by the height to get r², then square-root.

What are two ways to find the volume of a hollow cylinder?

Area of the ring-shaped cross-section × height, or the big cylinder's volume − the inner cylinder's volume.

In V = πr²h, what's the difference between height, length and depth?

Nothing. They all mean how far the circle is stacked.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Which solids stack like a cylinder?

Four ways to read πr²h

Find the diameter from the volume

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