Surface Area and Volume of a Sphere – Formulas and Examples

A sphere is the simplest 3D shape -- every point on its surface is the same distance from the centre. It is described by just one parameter: the radius $r$ .

What is a sphere?
Key dimensions
Surface area
Volume
Worked example 1 -- Surface area
Worked example 2 -- Volume
Worked example 3 -- Finding the radius from volume
Fun facts about spheres
Summary of formulas
Practice exercises

What is a sphere?

A sphere is the set of all points in 3D space that are at a fixed distance (the radius) from a central point (the centre).

         ___──────___
       ╱╱             ╲╲
      ╱       ●──r      ╲       ● = centre
     │                    │
      ╲                  ╱
       ╲╲___       ___╱╱
            ───────

Key characteristics:

A sphere has no edges and no vertices
It has one continuous curved surface
It is perfectly symmetrical in every direction
It is uniquely defined by its radius $r$ (or equivalently its diameter $d = 2 r$ )

A sphere has no flat faces, so there is no base area or lateral area to worry about -- there is only the total surface area.

Key dimensions

Symbol	Meaning
$r$	Radius -- distance from the centre to any point on the surface
$d$	Diameter -- $d = 2 r$

That is all you need. Unlike cylinders, cones, and pyramids, the sphere requires no height and no slant height.

Surface area

The surface area of a sphere is:

S = 4 π r^{2}

Interesting connection: The surface area of a sphere equals exactly 4 times the area of a circle with the same radius: $S = 4 \cdot π r^{2}$ .

If you know the diameter instead:

S = 4 π (\frac{d}{2})^{2} = π d^{2}

Volume

The volume of a sphere is:

V = \frac{4}{3} π r^{3}

If you know the diameter:

V = \frac{4}{3} π (\frac{d}{2})^{3} = \frac{π d ^{3}}{6}

Memory trick: The volume formula has $r^{3}$ (cubic, since volume is 3D) and the fraction $\frac{4}{3}$ .

Worked example 1 -- Surface area

Problem: A sphere has radius

r = 10 cm

. Find its surface area. Solution:

S = 4 π r^{2} = 4 π \cdot 1 0^{2} = 4 π \cdot 100 = 400 π

S \approx 1 256.64 cm^{2}

Worked example 2 -- Volume

Problem: A basketball has a diameter of

d = 24 cm

. Find its volume. Solution:

First, find the radius: $r = \frac{d}{2} = 12 cm$ .

V = \frac{4}{3} π r^{3} = \frac{4}{3} π \cdot 1 2^{3} = \frac{4}{3} π \cdot 1728

V = \frac{6 912}{3} π = 2304 π

V \approx 7 238.23 cm^{3} \approx 7.24 l

Worked example 3 -- Finding the radius from volume

Problem: A spherical balloon has a volume of

V = 5000 cm^{3}

. What is its radius? Solution:

Start from the volume formula and solve for $r$ :

V = \frac{4}{3} π r^{3}

r^{3} = \frac{3 V}{4 π} = \frac{3 \cdot 5 000}{4 π} = \frac{15 000}{4 π} = \frac{3 750}{π}

r^{3} \approx 1 193.66

r = 3 1 193.66

r \approx 10.61 cm

Fun facts about spheres

Optimal shape: Of all solids with a given volume, the sphere has the smallest surface area. This is why soap bubbles are spherical -- nature minimises surface tension.

Earth as a sphere: The Earth is roughly spherical with a mean radius of about $6371 km$ . Its surface area is approximately $5.1 \times 1 0^{8} km^{2}$ and its volume is about $1.08 \times 1 0^{12} km^{3}$ .

Archimedes' discovery: Archimedes proved that a sphere inscribed in a cylinder (same radius and height $= 2 r$ ) has exactly $\frac{2}{3}$ of the cylinder's volume and $\frac{2}{3}$ of its total surface area. He considered this his greatest achievement.

Doubling the radius: If you double the radius of a sphere, the surface area increases by a factor of $4$ (since $S \propto r^{2}$ ) and the volume increases by a factor of $8$ (since $V \propto r^{3}$ ).

Summary of formulas

Quantity	Formula
Surface area	$S = 4 π r^{2}$
Volume	$V = \frac{4}{3} π r^{3}$

Practice exercises

Put your knowledge to the test: