A sphere: surface area and volume

The most perfect solid

A ball, a bubble, a planet. A sphere is a solid with no edges, faces or vertices — just one centre and one radius r. Every point of the surface is equally far from the centre.

That is why it is the simplest for calculations in the question: one number is enough. But this time you have to remember the formulae — you will not derive them from a prism or a cylinder by a simple division.

Volume of a sphere: V = (4/3) · π · r³. Notice the cube: r³ = r · r · r.

Surface of a sphere: S = 4 · π · r². Here the power is two.

A helper so they do not mix: volume is always “cubic” (cm³ → a cube), surface “square” (cm² → a square). The power in the formula matches the unit of the result.

That three in the power has a consequence worth remembering: twice the radius means eight times the volume. A small growth of a ball, a huge growth of the contents. Today you will calculate it in black and white.

Adam and Sofie are picking a beach ball in a shop. The smaller has diameter 30 cm, the larger 60 cm and it is three times more expensive. “A twice bigger ball for a triple price? A rip-off,” Adam scowls. Sofie counts: “Wait. Radii 15 and 30 cm. Volume grows with the cube: twice the radius is two to the third — eight times more air.” Adam blinks: “Eight times more ball for a triple price. I’ll take the large one.” A cube can surprise — both ways.

💡

Calculate a cube step by step and write middle results: 6³ = 6 · 6 · 6 = 36 · 6 = 216. Who calculates r³ from the head in one go makes needless mistakes.

Volume and surface of a sphere step by step

Volume:

Step 1: Find the radius. From the diameter: r = diameter / 2. Step 2: The cube: r³ = r · r · r. Write a middle result. Step 3: Substitute: V = (4/3) · π · r³. Practically: r³ times π, times 4, divided by 3.

Example: r = 3 cm. r³ = 27. V = (4/3) · 3.14 · 27 = 4 · 3.14 · 9 = 113.04 cm³. (A trick: 27/3 = 9, do the dividing by three at once so the numbers stay small.)

Surface:

Step 1: Radius r. Step 2: Square r². Step 3: S = 4 · π · r².

Example: r = 3 cm. r² = 9. S = 4 · 3.14 · 9 = 113.04 cm².

Yes, for r = 3 the volume and the surface come out the same as a number — that is a coincidence of this three, not a rule. The units differ: cm³ against cm².

A hemisphere (a dome, a bowl): volume is half a sphere. Surface watch — half a sphere (2 · π · r²) plus a circular lid (π · r²), if the solid has one. Again: read the question and think which surfaces are really there.

Example 1: the volume of a ball

Question: A football has diameter 22 cm (rounded). How many litres of air does it contain? (1 litre = 1000 cm³)

Method: Radius: r = 22/2 = 11 cm.

Cube: 11³ = 11 · 11 · 11 = 121 · 11 = 1331.

Volume: V = (4/3) · 3.14 · 1331 ≈ 4 · 3.14 · 443.7 ≈ 5572.8 cm³.

Convert: 5572.8 cm³ ≈ 5.6 litres.

Answer: There is about 5.6 litres of air in the ball — more than two large bottles of lemonade.

Example 2: the surface of a globe

Question: A globe has radius 20 cm. How many cm² of paper with a map is needed to cover it?

Method: Covering = surface.

r² = 400.

S = 4 · 3.14 · 400 = 5024 cm².

Sense check: 5024 cm² is about half a square metre — believable for a globe of diameter 40 cm.

Answer: About 5024 cm² is needed for covering, so about 0.5 m² of map paper.

Example 3: twice the radius

Question: A smaller sphere has r = 2 cm, a larger r = 4 cm. How many times does the larger sphere have a larger volume?

Method: Smaller: r³ = 8, V = (4/3) · 3.14 · 8 ≈ 33.49 cm³.

Larger: r³ = 64, V = (4/3) · 3.14 · 64 ≈ 267.95 cm³.

Ratio: 267.95/33.49 = 8.

Answer: Eight times. Double the radius = 2³ = 8 times the volume. The surface would grow “only” four times (2²). That is why a large pack is often better value — the contents grow faster than the wrapping.

⚠️

r³ is not 3 · r. The most common sphere mistake: instead of 5³ = 125 writing 15. A cube means multiply the number by itself three times, not multiply by three. Write a middle step: 5³ = 5 · 5 · 5 = 25 · 5 = 125. Three extra seconds, the whole question saved.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Practise the area of a circle (π · r²) — the surface of a sphere is exactly four times that. Calculate with π ≈ 3.14, round to one decimal place.

Practise the questions

On paper

Write powers with middle results, π ≈ 3.14, units with every result.

A sphere with radius 6 cm: calculate the volume. (Hint: 6³ = 216 and 216/3 = 72.)
A sphere with diameter 10 cm: calculate the surface. Radius first!
A hemisphere (a dome) with radius 3 m: how many m² of sheet for covering it? (Only the round part, without the base.)
Compare: a sphere with r = 3 cm and a sphere with r = 9 cm. How many times more volume does the larger have? Estimate through the cube (3³) and check by calculation.

Now you 💪

You know both formulae and you know which has a square and which a cube (by the units of the result).
You calculate a cube step by step with middle results.
You know that double the radius = eight times the volume, and you can explain why.

✅

Done when: You have calculated the volume and the surface of a sphere, the surface of a hemisphere and checked the eightfold jump of volume at double the radius.

What to take from this lesson

Volume of a sphere: V = (4/3) · π · r³. Surface: S = 4 · π · r².
The power matches the unit: a cube for cm³, a square for cm².
r³ = r · r · r, write middle results.
Double the radius = 8 times larger volume, 4 times larger surface.
For a hemisphere, think whether the circular lid is counted too.