# How is the surface of a sphere calculated

### The earth is a sphere

Many objects are spherical. Or almost spherical: you can approach planet earth with a sphere, although it is "flatter" at the north and south poles.

Image: iStockphoto.com (evirgen)

How big is the surface of the earth?

To calculate the surface area, you need a few properties of the sphere.

The sphere has a center. All points on the surface are equidistant from the center. The distance is the radius or spherical radius.

A sphere is a geometric body that you get when you rotate a circle around its diameter.

### What is the surface of a sphere?

The surface of a sphere consists of a curved surface. The best way to imagine it would be to cut the ball open. Imagine you cut the surface of the sphere into many strips.

This is how you calculate the surface area of ​​a sphere: \$\$ O = pi * 4 * r ^ 2 \$\$ or \$\$ O = pi * d ^ 2 \$\$

• \$\$ d \$\$ diameter
• \$\$ pi \$\$ circle number

The surface is the area that you can touch when you hold the ball in your hand.

\$\$ O = pi * 4 * r ^ 2 = pi * d ^ 2 \$\$, because \$\$ 2 * r = d \$\$ and thus \$\$ (2 * r) ^ 2 = 4 * r ^ 2 = d ^ 2 \$\$

### Calculate the surface of a sphere

Given is a sphere with \$\$ d = \$\$ \$\$ 8 \$\$ \$\$ cm \$\$.

You know:

\$\$ d = 2 * r \$\$

\$\$ d / 2 = r \$\$

\$\$ (8 cm) / 2 = r \$\$

\$\$ 4 cm = r \$\$

To calculate the surface of the sphere, do the following:

\$\$ O = pi * 4 * r ^ 2 \$\$

\$\$ O = pi * 4 * (4 cm) ^ 2 \$\$

\$\$ O = pi * 4 * 16 cm ^ 2 \$\$

\$\$ O = 201.06 cm ^ 2 \$\$

Or:

\$\$ O = pi * d ^ 2 \$\$

\$\$ O = pi * (8 cm) ^ 2 \$\$

\$\$ O = pi * 64 cm ^ 2 \$\$

\$\$ O = 201.06 cm ^ 2 \$\$

Use the \$\$ pi \$\$ key on your calculator.

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### Calculate the radius for a given surface

A sphere with a surface area \$\$ O = 453.54 cm ^ 2 \$\$ is given.

To calculate the radius of the sphere, do the following:

1. Plug the given surface area into the formula:

\$\$ O = pi * 4 * r ^ 2 \$\$

\$\$ 453.54 cm ^ 2 \$\$\$\$ = pi * 4 * r ^ 2 \$\$

2 .. Solve the formula for \$\$ r \$\$:

\$\$ 453.54 cm ^ 2 \$\$\$\$ = pi * 4 * r ^ 2 \$\$ | \$\$: 4 \$\$ | \$\$: pi \$\$

\$\$ (453.54 cm ^ 2) / (pi * 4) \$\$\$\$ = r ^ 2 \$\$ | \$\$ sqrt \$\$

\$\$ sqrt ((453.54 cm ^ 2) / (pi * 4)) \$\$\$\$ = r \$\$

\$\$ sqrt (36.09 cm ^ 2) = r \$\$

\$\$ 6.01 cm = r \$\$

You can also first solve the formula for \$\$ r \$\$ and then insert the given surface:

\$\$ O = pi * 4 * r ^ 2 \$\$ | \$\$: 4 \$\$ | \$\$: pi \$\$

\$\$ O / (pi * 4) = r ^ 2 \$\$ | \$\$ sqrt \$\$

\$\$ sqrt (O / (pi * 4)) = r \$\$

\$\$ sqrt ((453.54 cm ^ 2) / (pi * 4)) \$\$\$\$ = r \$\$

\$\$ sqrt (36.09 cm ^ 2) = r \$\$

\$\$ 6.01 cm = r \$\$

### Calculate the diameter for a given surface

Given is a sphere with a surface area of ​​\$\$ O = 453.54 \$\$ \$\$ cm ^ 2 \$\$.

To calculate the diameter of the sphere, do the following:

1. Plug the given surface area into the formula:

\$\$ O = pi * d ^ 2 \$\$

\$\$ = pi * d ^ 2 \$\$

2. Solve the formula for \$\$ d \$\$:

\$\$ 453.54 cm ^ 2 \$\$\$\$ = pi * d ^ 2 \$\$ | \$\$: pi \$\$

\$\$ (453.54 cm ^ 2) / pi = d ^ 2 \$\$ | \$\$ sqrt \$\$

\$\$ sqrt ((453.54 cm ^ 2) / pi) = d \$\$

\$\$ sqrt (144.37 cm ^ 2) = d \$\$

\$\$ 12.02 cm = d \$\$