A cylinder is a prism with a circular end, so its volume is V = πr²h. A cone with the same base and height holds one third as much, so its volume is V = ⅓πr²h.
Both appear with the capacity questions in volume and capacity. This lesson builds directly on calculating a prism volume.
How do the formulae connect?
The end of a cylinder is a circle of area πr². Multiply by the length, here called the height h, and you have the prism rule again: πr² × h.
The cone adds one idea. It narrows to a point, so it needs a third of the cylinder’s volume. The height in the formula is the vertical height, from the centre of the base straight up to the tip.
How to work through a question
- Write down r and h. If the question gives a diameter, halve it.
- Check the height is vertical. If you are given a slant height l, find h using Pythagoras: h² = l² − r².
- Substitute into the correct formula.
- Calculate with the π button and round only at the end.
- State the unit, cubic.
Worked example
A cone has base radius 6 cm and a slant height of 10 cm. Find its volume, to 3 significant figures.
Step 1, find the vertical height: h² = 10² − 6² = 100 − 36 = 64, so h = 8 cm.
Step 2, substitute: V = ⅓ × π × 6² × 8 = ⅓ × π × 36 × 8 = 96π.
Step 3, calculate: 96π = 301.59…, so 302 cm³ to 3 significant figures.
The mistake to watch for
A frequent error is to use the slant height in place of the vertical height.
Mistaken working: V = ⅓ × π × 6² × 10 = 120π = 377 cm³
The 10 cm runs along the sloping side, so it is not the height.
The correction is to ask “which length runs straight up from the centre of the base?” Here it is 8 cm, found with Pythagoras, so the volume is 302 cm³. Notice that the wrong answer is larger than it should be, because the slant is always longer than the height.
Check yourself
Give answers to 3 significant figures.
1. A cylinder has radius 4 cm and height 7.5 cm. Find its volume.
Show answer
V = π × 4² × 7.5 = π × 16 × 7.5 = 120π = 376.99… so 377 cm³.
2. A cone has base radius 3 cm and vertical height 12 cm. Find its volume.
Show answer
V = ⅓ × π × 3² × 12 = ⅓ × π × 9 × 12 = 36π = 113.09… so 113 cm³.
3. A cylindrical tin has diameter 10 cm and height 3 cm. Find its volume.
Show answer
The radius is 10 ÷ 2 = 5 cm. V = π × 5² × 3 = 75π = 235.61… so 236 cm³.
Where this leads next
Next, convert between volume units and capacity, because cylinders such as tins and tanks are usually described in litres. The non-calculator working trainer can help when a question asks for an exact answer in terms of π.
Students often know the formulae but choose the wrong length under pressure. Seeing that happen live is one reason people look into online one-to-one Mathematics tuition.