Surface Area & Volume of Spheres

Q: How do I remember which formula squares and which cubes?

Surface area is a 2D measure, so it uses $r^2$. Volume fills 3D space, so it uses $r^3$. The power always matches the dimension.

Q: How is a hemisphere's surface area different?

It is $3\pi r^2$ — half the sphere's curved surface ($2\pi r^2$) plus the flat circular base ($\pi r^2$). Do not forget the cap.

A sphere is the roundest solid there is, and it needs only one measurement: its radius. Both its surface area and its volume come from formulas built entirely from $r$ .

Surface area is $4\pi r^2$ and volume is $\dfrac{4}{3}\pi r^3$ . The only real work is squaring or cubing the radius correctly and keeping the constants straight.

The two formulas

Surface area of a sphere is $4\pi r^2$ — exactly four times the area of a flat circle with the same radius. Volume is $\dfrac{4}{3}\pi r^3$ .

Notice surface area squares the radius (it measures a 2D surface) while volume cubes it (it fills 3D space). Matching the power to the dimension is a good check.

Hemispheres and common traps

A hemisphere is half a sphere, so its volume is half of $\dfrac{4}{3}\pi r^3$ . Its surface area, though, is half the sphere plus the flat circular cap: $2\pi r^2 + \pi r^2 = 3\pi r^2$ . The figure shows the hemisphere with that flat circular face on top.

The usual mistake is using diameter instead of radius. If a problem gives the diameter, cut it in half before plugging in.

Worked examples

Example 1: volume of a sphere

Find the volume of a sphere with radius $6$ .

Volume formula

V = \tfrac{4}{3}\pi r^3

Cube the radius

6^3 = 216

Multiply

\tfrac{4}{3}\pi(216) = 288\pi

Answer: $288\pi$ cubic units

Example 2: surface area of a sphere

Find the surface area of a sphere with radius $5$ .

Surface area formula

S = 4\pi r^2

Square the radius

5^2 = 25

Multiply

4\pi(25) = 100\pi

Answer: $100\pi$ square units

Example 3: surface area of a hemisphere from its diameter

Find the surface area of a hemisphere with diameter $8$ .

Halve the diameter to get the radius

r = \tfrac{8}{2} = 4

Half the curved surface plus the flat cap

S = 2\pi r^2 + \pi r^2 = 3\pi r^2

Square the radius

4^2 = 16

Multiply

3\pi(16) = 48\pi

Answer: $48\pi$ square units

Try one yourself

Common questions

How do I remember which formula squares and which cubes?

Surface area is a 2D measure, so it uses $r^2$ . Volume fills 3D space, so it uses $r^3$ . The power always matches the dimension.

What if I am given the diameter?

Halve it to get the radius before using either formula. A diameter of $12$ means $r = 6$ .

How is a hemisphere's surface area different?

It is $3\pi r^2$ — half the sphere's curved surface ( $2\pi r^2$ ) plus the flat circular base ( $\pi r^2$ ). Do not forget the cap.

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Surface Area & Volume of Spheres

The two formulas

Hemispheres and common traps

Worked examples

Example 1: volume of a sphere

Example 2: surface area of a sphere

Example 3: surface area of a hemisphere from its diameter

Try one yourself

Common questions

Keep learning

Volume of Pyramids & Cones

Surface Area of Prisms & Cylinders

Scale Factor: Perimeter, Area & Volume