Surface Area and Volume • Topic 4 of 7

Cones

A cone has a circular base and tapers to a single point (the apex). Its volume is one-third that of a cylinder with the same base and height: ⅓πr²h. The factor of ⅓ is the key feature — forgetting it gives three times the correct answer. For radius 3 and height 4, the volume is ⅓π(9)(4) = 12π. Compute πr²h first, then take a third, and leave the answer in terms of π. As with cylinders, halve a given diameter to get the radius. Cones appear in ice-cream, funnel and pile problems. Remember the same ⅓ factor also applies to pyramids.

A cone with radius, height and apexConehrVolume = ⅓πr²h

✅ Solved examples

1. A cone has radius 3 and height 4. Find its volume in terms of π.
⅓π(9)(4) = 12π.
2. A cone has radius 3 and height 6. Find its volume.
⅓π(9)(6) = 18π.
3. A cone has radius 6 and height 5. Find its volume.
⅓π(36)(5) = 60π.
4. A cone has radius 2 and height 9. Find its volume.
⅓π(4)(9) = 12π.

✏️ Practice — try these, take hints as needed

1. A cone has radius 3 and height 5. Find its volume in terms of π.
⅓πr²h.
⅓π(9)(5).
—
15π.
2. A cone has radius 6 and height 4. Find its volume.
⅓π(36)(4).
—
—
48π.
3. A cone has radius 3 and height 9. Find its volume.
⅓π(9)(9).
—
—
27π.
4. A cone has radius 9 and height 2. Find its volume.
⅓π(81)(2).
—
—
54π.
5. A cone has radius 2 and height 6. Find its volume.
⅓π(4)(6).
—
—
8π.

📝 Topic test — 8 questions

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