Surface Area and Volume of a Sphere

The surface area of a sphere is the number of square units that will exactly cover its surface.

surface area and volume of a sphere

The formula for the surface area of a sphere is given by:

A = 4r²π

where A represents the area and r represents the radius.

If you already know the surface area and need to find the radius, we can rearrange the above formula and solve it for r:

 A = 4r^2\pi

\frac{A}{4\pi} = r^2

and now take the square root of both sides:


divide both sides with 4π

The volume of a sphere is the number of cubic units that will exactly fill the sphere.

Volume of a sphere is given by:


where V represents the volume and r is the radius.

Again, we can solve the above formula for r and thus find the radius if we are given the volume:



and now take the third root of both sides


divide both sides with \frac{4}{3}\pi

Interesting fact

The shape with the smallest possible surface area for a given volume is a sphere. Or in other words, the sphere is a shape with the largest volume for a fixed surface area.

The sphere therefore appears in nature in water drops, bubbles, planets etc.

Example I

Find the surface area and the volume of a sphere with radius 5.

To do this, simply plug in r=5 into the formulas.

For the area:



A = 4 * 125 * \pi

A = 500\pi

We can plug in 3.14π for and get:


A = 1570

For the volume:

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

V=\frac{4}{3}\ 5^{3}\pi



Again, we can plug in 3.14 for π and get:



Example II

Find the radius of a sphere with a volume of 65.42.

Now we’ll use the last formula provided above, the one solved for r. Plug in 65.42 for V and 3.14 for π:





More Examples

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