Volume and Surface Area of a Sphere

Use V = 4/3 πr³ and A = 4πr² for spheres and hemispheres, find the radius from a surface area, work out hollow shapes as big minus small, and compare two spheres.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1 · 22 real past-paper questions below

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Two Formulas, One Radius

A sphere is a perfect ball. Everything about it depends on its radius. The exam gives you both formulas.

Formula

V = \frac{4}{3} π r^{3} A = 4 π r^{2}

$V$ is the volume, in cm³. $A$ is the surface area, in cm². $r$ is the radius: half the diameter.

Worked example

1
$d = 12, r = 12 \div 2 = 6$ — The ball is 12 cm across: that is the diameter. Halve it, because both formulas need the radius.
2
$6^{3} = 6 \times 6 \times 6 = 216$ — Volume first. Cube the radius before anything else: multiply it by itself three times.
3
$V = \frac{4}{3} π \times 216$ — Put 216 in place of $r^{3}$ in the volume formula.
4
$= 288 π$ — 216 ÷ 3 = 72, and 72 × 4 = 288. If the question says “in terms of π”, stop here.
5
$\approx 904.8 cm^{3}$ — Press the π key: 288 × π = 904.77…. Volume is in cm³, because three lengths are multiplied.
6
$6^{2} = 6 \times 6 = 36$ — Now the surface. Square the same radius.
7
$A = 4 π \times 36 = 144 π$ — Put 36 in place of $r^{2}$ , and 4 × 36 = 144.
8
$\approx 452.4 cm^{2}$ — 144 × π = 452.38…. Area is in cm², because two lengths are multiplied.

Common mistake

Using the diameter. If the question gives 12 cm across, halve it first: $r = 6$ .

Key idea

A hemisphere is half a sphere. Its volume is half. Its total surface is the curved half, $2 π r^{2}$ , plus the flat circle, $π r^{2}$ : $3 π r^{2}$ .

Key idea

Archimedes, in Sicily around 250 BC, proved that a sphere fills exactly $\frac{2}{3}$ of the cylinder that fits snugly round it. He is said to have asked for a sphere and cylinder on his tomb, and about 140 years later the Roman writer Cicero said he found it by that carving.

From Surface Area to Volume

The 3-mark question: you are told the surface area. Find the radius first, then the volume.

A sphere has surface area $100 π cm^{2}$ . Find its volume.

Worked example

1
$4 π r^{2} = 100 π$ — Write the surface area formula equal to the number you are given. Only $r$ is unknown.
2
$r^{2} = \frac{100 π}{4 π} = 25$ — Divide both sides by 4π to get $r^{2}$ on its own. The π's cancel, and 100 ÷ 4 = 25.
3
$r = 25 = 5$ — Square root to undo the square. The radius is 5. Use the positive root: a radius cannot be negative.
4
$V = \frac{4}{3} π \times 5^{3}$ — Now put the radius into the volume formula.
5
$= \frac{4}{3} π \times 125 = \frac{500}{3} π$ — 5³ = 125, and 4 × 125 = 500.
6
$\approx 523.6 \approx 524 cm^{3}$ — 500 ÷ 3 × π = 523.59…, so 524 to the nearest whole number.

Common mistake

Stopping at 25. That is r squared, so take the square root.

From Volume to Radius

Exams also give the volume and ask for the radius. Undo the formula one step at a time. Here r is cubed, so the last step is a cube root.

A sphere has volume $972 π cm^{3}$ . Find its radius.

Worked example

1
$\frac{4}{3} π r^{3} = 972 π$ — Set the volume formula equal to the volume you are given.
2
$4 π r^{3} = 2916 π$ — Times both sides by 3 to clear the fraction: 972 × 3 = 2916.
3
$r^{3} = \frac{2916 π}{4 π} = 729$ — Divide both sides by 4π. The π's cancel, and 2916 ÷ 4 = 729.
4
$r = 3729 = 9 cm$ — Cube root to undo the cube: 9 × 9 × 9 = 729.

A hemisphere has volume 200 cm³. Find its radius.

Worked example

1
$\frac{2}{3} π r^{3} = 200$ — A hemisphere is half a sphere, so its volume is half of $\frac{4}{3} π r^{3}$ , which is $\frac{2}{3} π r^{3}$ .
2
$2 π r^{3} = 600$ — Times both sides by 3: 200 × 3 = 600.
3
$r^{3} = \frac{600}{2 π} = 95.49 \dots$ — Divide both sides by 2π. Keep every digit on the calculator.
4
$r = 3 95.49 \dots \approx 4.57 cm$ — Cube root to undo the cube. 4.57 to 3 significant figures.

Common mistake

Taking the square root at the end. r is cubed, so you need the cube root: √729 = 27 is wrong, ∛729 = 9 is right.

Key idea

Show the line with $r^{3}$ on its own. Most of the marks are for the working, so a radius found by guessing scores little.

Hollow Balls and Bowls

The 5-mark question: a hollow ball or a bowl with thick walls. The metal is the big solid take away the small one inside.

A bowl is a hemisphere. The outside radius is 6 cm and the metal is 1 cm thick.

Worked example

1
$inside radius = 6 - 1 = 5$ — The metal is 1 cm thick, so the hollow inside is 1 cm smaller than the outside.
2
$\frac{4}{3} π \times 6^{3} = 288 π$ — Volume of the whole outside ball, as if it were solid: 6³ = 216, and 4/3 of 216 is 288.
3
$\frac{4}{3} π \times 5^{3} = \frac{500}{3} π$ — Volume of the empty space inside: 5³ = 125, and 4 × 125 = 500.
4
$288 π - \frac{500}{3} π = \frac{364}{3} π$ — Metal = big minus small. Write 288 as 864/3, then 864 − 500 = 364.
5
$\frac{364}{3} π \div 2 = \frac{182}{3} π$ — The bowl is a hemisphere, so halve it.
6
$\approx 190.6 cm^{3}$ — 182 ÷ 3 × π = 190.59…. This is the volume of metal.

Common mistake

Working out $(6 - 5)^{3} = 1$ . Cube each radius first, then subtract.

Key idea

If the metal has a density of 8 grams per cm³, the mass is $190.6 \times 8 \approx 1525$ grams. See Compound Measures.

Sphere Inside a Cube

A ball that fits exactly inside a box touches all six faces. So the sphere's diameter is the side of the cube.

A cube has volume 1000 cm³. A sphere fits exactly inside it. What percentage of the cube is empty space?

Worked example

1
$side = 31000 = 10 cm$ — The cube's volume is side³, so cube root it to get the side.
2
$r = 10 \div 2 = 5$ — The sphere touches opposite faces, so its diameter is 10. Halve it: the radius is 5.
3
$V = \frac{4}{3} π \times 5^{3} = \frac{500}{3} π$ — Sphere volume: 5³ = 125, and 4 × 125 = 500.
4
$\approx 523.6 cm^{3}$ — 500 ÷ 3 × π = 523.59…. Keep the full value on the calculator for the next line.
5
$1000 - 523.6 = 476.4 cm^{3}$ — The empty space is the cube take away the sphere.
6
$\frac{476.4}{1000} \times 100 = 47.6%$ — Percentage = empty space ÷ whole cube × 100.

Common mistake

Using the side 10 as the radius. The side goes right across the ball, so it is the diameter.

Key idea

The answer is always about 47.6%, for any size of cube. A sphere fills $\frac{π}{6} \approx 0.524$ of the cube it fits in.

Melted and Recast

When a solid is melted and made into a new shape, no metal is lost. So the two volumes are equal. Write that as an equation.

A solid metal sphere of radius 6 cm is melted and made into a cylinder of radius 4 cm. Find the height of the cylinder.

Worked example

1
$\frac{4}{3} π \times 6^{3} = 288 π$ — Volume of the sphere: 6³ = 216, and 4/3 of 216 is 288. Keep π, because it will cancel.
2
$π \times 4^{2} \times h = 288 π$ — The cylinder is made from the same metal, so its volume $π r^{2} h$ equals 288π.
3
$16 π h = 288 π$ — Square the cylinder's radius: 4² = 16.
4
$h = \frac{288 π}{16 π} = 18 cm$ — Divide both sides by 16π. The π's cancel, and 288 ÷ 16 = 18.

Key idea

Making small balls instead? Divide the big volume by one small ball's volume. A ball of radius 2 holds $\frac{32}{3} π$ , and $288 π \div \frac{32}{3} π = 27$ balls.

Key idea

A ball dropped into a tank of water pushes the water up by its own volume. So the ball's volume = $π r^{2} \times$ the rise, where r is the tank's radius.

Two Spheres (Extended/Higher)

The 5-mark question compares two spheres. If one surface area is 4 times the other, its radius is 2 times, and its volume is 8 times.

The difference in their volumes is $252 π cm^{3}$ . Find the smaller radius.

Worked example

1
$radii r and 2 r$ — Area grows with the square of the radius. Areas 4 times means radius √4 = 2 times.
2
$\frac{4}{3} π r^{3} and \frac{4}{3} π (2 r)^{3}$ — Write each volume with its own radius.
3
$(2 r)^{3} = 8 r^{3}$ — Cube the 2 as well as the r: 2 × 2 × 2 = 8.
4
$\frac{4}{3} π (8 r^{3} - r^{3}) = 252 π$ — The difference is big minus small. Set it equal to 252π.
5
$\frac{4}{3} π \times 7 r^{3} = 252 π$ — 8r³ − r³ = 7r³.
6
$\frac{28}{3} r^{3} = 252$ — Divide both sides by π, and 4/3 × 7 = 28/3.
7
$r^{3} = 252 \times \frac{3}{28} = 27$ — Times both sides by 3, then divide by 28.
8
$r = 327 = 3 cm$ — Cube root: 3 × 3 × 3 = 27. This is the smaller radius.

Key idea

Areas grow with the square of the scale, volumes with the cube. See Volume Factor of Similar Shapes.

Which Method? Read the Question

Find what you are given, then what you are asked for.

You see	You do
Radius, find volume or area	put the radius in the formula
Surface area given	undo 4π, square root, then volume
Volume given, find the radius	undo $\frac{4}{3} π$ , then cube root
Hollow, bowl, thick metal	big minus small
Hemisphere	half the volume; surface $3 π r^{2}$
Two spheres, areas 4 times	radius 2 times, volume 8 times
Sphere fits exactly in a cube	diameter = side of the cube
Melted and made into a new shape	set the two volumes equal
Cones and pointed tops	Volume and Surface Area of a Cone

Key idea

Leave answers in terms of π until the very end. Round only once.

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