Maths
How do you find the volume of a cone and a sphere?
Quick answer
The volume of a cone is V = ⅓πr²h and the volume of a sphere is V = ⁴⁄₃πr³, where r is the radius and h the height. Answers are in cubic units, like cm³.
In one sentence
A cone holds one third of the cylinder it fits in, so V = ⅓πr²h, and a sphere's volume is V = ⁴⁄₃πr³.
Explained
Volume is the amount of space a solid takes up, measured in cubic units such as cm³ or m³ (see volume). For a cylinder it is easy: the area of the circle at the bottom times the height, πr²h. Cones and spheres are a little trickier, but each has a short formula.
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Volume of a cone: V = ⅓πr²h
Here r is the radius of the circular base and h is the perpendicular height (straight up from the centre of the base to the tip). The formula says: work out the area of the circle at the base, πr², multiply by the height, then divide by 3. Why a third? If you fill a cone with water and pour it into a cylinder with the same base and height, it takes exactly three cones to fill the cylinder. A pyramid works the same way: one third of the prism around it.
Volume of a sphere: V = ⁴⁄₃πr³
Here r is the radius, the distance from the centre to the surface. The Greek mathematician Archimedes discovered that a sphere fills exactly two thirds of the cylinder that just fits around it (radius r, height 2r). That cylinder has volume πr² × 2r = 2πr³, and two thirds of 2πr³ is ⁴⁄₃πr³. A hemisphere (half a sphere) is half of that: ⅔πr³.
There is a lovely link between the three shapes. Take a cone, a sphere and a cylinder, all with radius r and height 2r. Their volumes are ⅔πr³, ⁴⁄₃πr³ and 2πr³: in the ratio 1 : 2 : 3.
Two things to watch. Both formulas use the radius, so if you are given the diameter, halve it first. And because the sphere formula cubes the radius, a small change in r makes a big change in volume: double the radius and the volume is 2³ = 8 times bigger. You can leave answers "in terms of π" (like 36π cm³), which is exact, or use a calculator and round (113.1 cm³).
Examples
| Shape | Working | Volume |
|---|---|---|
| Cone, r = 3 cm, h = 10 cm | ⅓ × π × 9 × 10 | 30π ≈ 94.2 cm³ |
| Cone, diameter 8 cm, h = 6 cm | r = 4; ⅓ × π × 16 × 6 | 32π ≈ 100.5 cm³ |
| Sphere, r = 5 cm | ⁴⁄₃ × π × 125 | ⁵⁰⁰⁄₃π ≈ 523.6 cm³ |
| Sphere, diameter 6 cm | r = 3; ⁴⁄₃ × π × 27 | 36π ≈ 113.1 cm³ |
| Hemisphere, r = 6 cm | ⅔ × π × 216 | 144π ≈ 452.4 cm³ |
Worked example with the slant height: a cone has radius 5 cm and slant height 13 cm. The slant height is the sloping side, not the height. The radius, the height and the slant height make a right-angled triangle, so by Pythagoras h = √(13² − 5²) = √144 = 12 cm. Then V = ⅓ × π × 25 × 12 = 100π ≈ 314.2 cm³.
Common mistakes
- Using the diameter instead of the radius. Fix: halve the diameter first. A ball 10 cm across has r = 5 cm.
- Using the slant height as the height. Fix: the height is perpendicular to the base. Use Pythagoras, h = √(l² − r²).
- Forgetting the ⅓ for a cone (that gives the cylinder's volume) or using ⁴⁄₃πr² for a sphere. Fix: a sphere's radius is cubed, r³.
- Wrong units. Volume is in cubic units: cm³ or m³, not cm². 1 cm³ = 1 ml.
Step by step
Step by step
- Find the radiusIf you are given the diameter, divide it by 2.
- Find the height (cones only)Use the perpendicular height. If you only have the slant height l, use h = √(l² − r²).
- Choose the formulaCone: V = ⅓πr²h. Sphere: V = ⁴⁄₃πr³. Hemisphere: V = ⅔πr³.
- Substitute and calculateSquare or cube the radius first, then multiply. Keep π as it is if you want an exact answer.
- Round and add unitsRound as the question asks (often 1 decimal place) and write cm³ or m³.
Working backwards: to find the radius of a sphere from its volume, rearrange to r³ = 3V ÷ (4π) and take the cube root. For a cone, h = 3V ÷ (πr²).
Practice questions
Volume of cones and spheres
Try each one, then check your answer.
- Find the volume of a cone with radius 3 cm and height 7 cm.
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⅓ × π × 9 × 7 = 21π ≈ 66.0 cm³ - Find the volume of a cone with radius 6 cm and height 5 cm.
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⅓ × π × 36 × 5 = 60π ≈ 188.5 cm³ - A cone has diameter 10 cm and height 9 cm. Find its volume.
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r = 5; ⅓ × π × 25 × 9 = 75π ≈ 235.6 cm³ - Find the volume of a sphere with radius 3 cm.
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⁴⁄₃ × π × 27 = 36π ≈ 113.1 cm³ - Find the volume of a sphere with radius 6 cm.
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⁴⁄₃ × π × 216 = 288π ≈ 904.8 cm³ - A sphere has diameter 10 cm. Find its volume.
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r = 5; ⁴⁄₃ × π × 125 ≈ 523.6 cm³ - Find the volume of a sphere with radius 1.5 m.
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⁴⁄₃ × π × 3.375 = 4.5π ≈ 14.1 m³ - Find the volume of a hemisphere with radius 3 cm.
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⅔ × π × 27 = 18π ≈ 56.5 cm³ - Give the volume of a cone with radius 2 cm and height 9 cm in terms of π.
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⅓ × π × 4 × 9 = 12π cm³ - A cone has radius 8 cm and slant height 10 cm. Find its volume.
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h = √(100 − 64) = 6; ⅓ × π × 64 × 6 = 128π ≈ 402.1 cm³ - A sphere has volume 36π cm³. Find its radius.
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⁴⁄₃r³ = 36, so r³ = 27 and r = 3 cm - A cone has volume 50π cm³ and radius 5 cm. Find its height.
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⅓ × 25 × h = 50, so 25h = 150 and h = 6 cm - Which is bigger: a cone with radius 3 cm and height 12 cm, or a sphere with radius 3 cm?
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Cone: ⅓ × π × 9 × 12 = 36π. Sphere: 36π. They are equal. - An ice-cream shape is a cone (radius 3 cm, height 10 cm) with a hemisphere of radius 3 cm on top. Find its total volume.
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30π + 18π = 48π ≈ 150.8 cm³ - A cone with radius 6 cm and height 12 cm has its top 6 cm cut off (a small cone of radius 3 cm). Find the volume that is left.
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144π − 18π = 126π ≈ 395.8 cm³
Word problems
Cones and spheres in real life
Try each one, then check your answer.
- A football has a diameter of about 22 cm. Roughly what is its volume?
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r = 11; ⁴⁄₃ × π × 1331 ≈ 5575 cm³ (about 5.6 litres) - A paper cone cup has radius 4 cm and height 9 cm. How much water does it hold?
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⅓ × π × 16 × 9 = 48π ≈ 150.8 cm³, about 151 ml - A pile of sand is a cone with radius 2 m and height 1.5 m. What is its volume?
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⅓ × π × 4 × 1.5 = 2π ≈ 6.3 m³ - A metal sphere of radius 6 cm is melted down and made into cones with radius 3 cm and height 4 cm. How many cones can be made?
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Sphere 288π, each cone 12π: 288 ÷ 12 = 24 cones - A hemispherical bowl has radius 12 cm. How many litres does it hold when full?
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⅔ × π × 1728 = 1152π ≈ 3619 cm³ ≈ 3.6 litres - A hollow ball has outer radius 5 cm and inner radius 4 cm. What volume of rubber is it made of?
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⁴⁄₃π(125 − 64) = ⁴⁄₃π × 61 ≈ 255.5 cm³
Questions people ask
What is the formula for the volume of a cone?
V = ⅓πr²h, where r is the radius of the base and h is the perpendicular height. It is one third of the volume of a cylinder with the same base and height.
How do you find the volume of a cone with the slant height?
First find the real height with Pythagoras: h = √(l² − r²), where l is the slant height. Then use V = ⅓πr²h. For r = 5 cm and l = 13 cm, h = 12 cm and V = 100π ≈ 314.2 cm³.
Why is the volume of a cone divided by 3?
Because a cone fills exactly one third of the cylinder with the same base and height. You can check it by pouring water or rice: three full cones fill the cylinder. It can be proved with calculus or by comparing slices.
How do you find the volume of a sphere with the diameter?
Halve the diameter to get the radius, then use V = ⁴⁄₃πr³. Or use the diameter directly: V = πd³ ÷ 6. A sphere with diameter 6 cm has volume 36π ≈ 113.1 cm³.
What is the volume of a sphere with radius 3?
⁴⁄₃ × π × 3³ = 36π, which is about 113.1 cubic units (113.1 cm³ if the radius is 3 cm).
What is the volume of a cone with the top cut off?
That shape is called a frustum. Find the volume of the whole cone and subtract the small cone that was cut off. There is also a formula: V = ⅓πh(R² + Rr + r²), where R and r are the two radii and h is the height of the frustum.
What is the volume of a hemisphere?
Half a sphere: V = ⅔πr³. A hemisphere of radius 3 cm has volume 18π ≈ 56.5 cm³.
What units is volume in?
Cubic units, because you multiply three lengths: cm³, m³ or mm³. For liquids, 1 cm³ = 1 ml and 1,000 cm³ = 1 litre.
Quick quiz
Volume of cones and spheres
Check what you know with 4 quick questions.
Question 1 of 4
A cone has radius 3 cm and height 4 cm. What is its volume?
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Related topics
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
3D shapes
3D shapes are solid shapes with length, width and height, like cubes, spheres, cylinders and pyramids. They have faces, edges and vertices.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Surface area
Surface area is the total area of all the outside faces of a 3D shape. For a cube with side a, it is 6a², because a cube has six equal square faces.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a trapezium (trapezoid)
Add the two parallel sides, multiply by the height and halve: A = ½(a + b)h. Parallel sides of 4 cm and 6 cm, 3 cm apart, give an area of 15 cm².