# What is the area of a semicircle

### Area of a circle

The formula for calculating the area $$ A $$ of a circle with the radius $$ r $$ is:

$$ A = pi r ^ 2 $$.

If you are interested in how to justify the formula, take a look at the next page.

$$ pi $$ is not a rational number. This means that it cannot be represented as a fraction and has an infinite number of places after the decimal point.

$$ pi $$ $$ approx $$ 3.14

### Area of a circle

Take a circle of any size and divide it into any number of equal parts (e.g. 16 parts).

Now cut the circular area into these 16 parts and place 15 of them next to each other so that they result in a figure that resembles a rectangle. (The 16th part is halved and placed on the left and right.)

The area of the resulting rectangle is calculated by length times width.

The length of this tinkered rectangle corresponds approximately to half the circumference of the circle ($$ 1 / 2u = 1/2 * 2pir $$).

The width corresponds approximately to the radius $$ r $$.

Accordingly, the following applies:

$$ A $$ $$ = $$ length times latitude

$$ A = 1/2 * 2pir * r $$

$$ A = pir ^ 2 $$

Area of a circle:

$$ A = pir ^ 2 $$

Note: If you don't have a calculator with the $$ pi $$ key, use $$ pi approx 3.14 $$.

Area of a circle:

$$ A = pir ^ 2 $$

Note: If you don't have a calculator with the $$ pi $$ key, use $$ pi approx 3.14 $$.

### Lines in a circle

In memory of:

The diameter is twice the radius.

$$ d = 2 * r $$

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### Calculation of the area for a given radius

Calculate the area of the clock you see in the picture. The radius of the clock is $$ 15 $$ $$ cm $$.

$$ A = pi * r ^ 2 $$

$$ A = pi * (15 cm) ^ 2 $$

$$ A = pi * 225 cm ^ 2 $$

$$ A approx 706.9 cm ^ 2 $$

The area of the clock is therefore approximately $ 706.9 cm ^ 2 $.

$$ A = pir ^ 2 $$

### Calculation of the area for a given diameter

In restaurants, you often get your drink with a cardboard coaster. The round coasters have a diameter of $$ d = 107 $$ $$ mm $$. Calculate the area.

In order to be able to use the area formula, you first need the radius.

$$ r = d / 2 = (107 mm) / 2 = 53.5 mm $$

Now you can calculate the area.

$$ A = pi r ^ 2 $$

$$ A = pi * (53.5 mm) ^ 2 $$

$$ A approx 8992 mm ^ 2 $$

The area of the beer mat is about $$ 8992 $$ $$ mm ^ 2 $$ or the equivalent of $$ 89.92 $$ $$ cm ^ 2 $$.

$$ A = pir ^ 2 $$

$$ r = d / 2 $$

Image: fotolia.com (contrastwerkstatt)

### Calculation of the radius and the diameter for a given area

The area of a Frisbee is given as $$ 530 $$ $$ cm ^ 2 $$. Calculate the radius and diameter of the frisbee disc.

$$ A = pir ^ 2 $$

$$ 530 cm ^ 2 = pir ^ 2 $$

$$ (530 cm ^ 2) / pi = r ^ 2 $$

$$ sqrt ((530 cm ^ 2) / pi) = r $$

$$ 13 cm approx r $$

The radius of the frisbee is about $$ 13 $$ $$ cm $$.

Since you know the diameter is twice the radius, you only have to take the result times two to calculate the diameter.

The diameter of the frisbee is therefore approximately $ 26 $ $ cm $.

$$ A = pi * r ^ 2 $$

$$ r = sqrt (A / pi) $$

$$ d = 2r $$

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### Area of a circular ring

The diameter of the CD shown is $ 12 $ $ $ cm $ and the radius is $ 6 $ $ $ cm $.

Image: Anders ARTig Werbung + Verlag GmbH

Calculate the area.

$$ A = pir ^ 2 $$

$$ A = pi (6 cm) ^ 2 $$

$$ A approx 113 cm ^ 2 $$

Now the CD has a hole. So we have to subtract the area of the hole from our result.

The diameter of the hole is $$ 1.5 $$ $$ cm $$. Accordingly, the radius of the hole is $ 0.75 $ $ cm $ and the area is about $ 1.77 $ $ cm ^ 2 $.

$$ A = pir ^ 2 $$

$$ A = pi (0.75 cm) ^ 2 $$

$$ A approx 1.77 cm ^ 2 $$

The actual surface area of the CD is therefore $$ 111.23 $$ $$ cm ^ 2 $$.

$$ 113 $$ $$ cm ^ 2 $$ $$ - $$ $$ 1.77 $$ $$ cm ^ 2 $$ $$ = $$ $$ 111.23 $$ $$ cm ^ 2 $$.

$$ A = pi * r ^ 2 $$

$$ r = d / 2 $$

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