Volume and surface area of a sphere

A sphere is the ball shape, and it is the simplest of the 3D shapes to describe: it only has one length, the radius $r$, the distance from the middle to the outside. That one number is all we need for both the volume and the surface area.

When do I use this?

Whenever a task asks how much fits inside a ball, or how much covers it. Volume is inside, surface area is outside, as explained in 3D shapes: volume and surface area. In the GCSE exam both sphere formulas are given to you in the question when you need them.

The volume of a sphere

We can find the volume of a sphere if we just have the radius. Then we use the formula:

$$V = \frac{4}{3} \times \pi \times r^3$$

Take a sphere with radius $2$ cm. Then

$$V = \frac{4}{3} \times \pi \times 2^3 = \frac{4}{3} \times \pi \times 8 = \frac{32}{3}\pi \approx 33.51 \text{ cm}^3$$

So about $33.51$ cubic centimetres fit inside it. In terms of $\pi$ the answer is $\frac{32}{3}\pi$ cm³.

The surface area of a sphere

The surface area of the sphere can be found with the formula

$$S = 4 \pi \times r^2$$

For the same sphere, with $r = 2$ cm:

$$S = 4 \pi \times 2^2 = 4 \pi \times 4 = 16\pi \approx 50.27 \text{ cm}^2$$

So about $50.27$ square centimetres of wrapping paper would cover the ball.

Cubed for volume, squared for surface area

Notice the pattern in the two formulas. The volume has $r^3$ and comes out in cm³. The surface area has $r^2$ and comes out in cm². That is not a coincidence: a volume is always three lengths multiplied together, and an area two. If you find yourself with $r^2$ in a volume, or $r^3$ in a surface area, you've picked up the wrong formula.

Common mistakes

• Mixing up the two formulas. Volume is $\frac{4}{3} \times \pi \times r^3$, surface area is $4 \pi \times r^2$. The cube goes with the volume and the square goes with the surface area.
• Using the diameter instead of the radius. Both formulas use the radius. If the task gives you the diameter, halve it first.
• Giving the answer in the wrong unit. Volume in cm³, surface area in cm². If the unit is wrong, we have probably calculated wrong somewhere.

Related

All the shapes together are in 3D shapes: volume and surface area. The circle's radius and $\pi$ are explained in circles. The cylinder and the cone use the same $\pi \times r^2$ for their circles, in volume and surface area of a cylinder and volume and surface area of a cone.