Area of a Circle with Circumference – Formulas and Examples

The circumference is the length of the outline of a circle, while the area of the circle is a measure of the space occupied by the circle. We can calculate the area of a circle using the circumference by determining the value of the radius from the circumference.

In this article, we will learn about two main methods that we can use to determine the area of a circle when we know the circumference. We will solve some practice problems to apply what we have learned.

GEOMETRY
Fórmula del área de un círculo con la circunferencia

Relevant for

Learning to calculate the area of a circle using the circumference.

See examples

GEOMETRY
Fórmula del área de un círculo con la circunferencia

Relevant for

Learning to calculate the area of a circle using the circumference.

See examples

How to find the area of a circle using the circumference?

We can find the area of a circle if we know the circumference using two main methods.

The first method is to determine the length of the radius from the circumference, and then use that length in the standard formula for the area of a circle. Therefore, we remember that the circumference can be written in the following way:

$latex C=2\pi r$

Therefore, to find the length of the radius, we can divide the circumference by 2π.

Then, we use the Area of a Circle Formula with the radius found. Remember that the formula for the area of a circle is:

$latex A=\pi r^2$

The second method consists in using a formula for the area of a circle in terms of the circumference, which we will find next.


Formula for the area of a circle using the circumference

We can derive a formula for the area of a circle in terms of the circumference. This will save us from having to find the length of the radius before calculating the area of the circle.

Therefore, we start with the standard formula for the area of a circle:

$latex A=\pi r^2$

Now, we can substitute the following equation:

$latex C=2\pi r$

$latex r=\frac{C}{2\pi}$

Therefore, we have:

$latex A=\pi r^2$

$latex A=\pi (\frac{C}{2\pi})^2$

$latex A=\pi (\frac{C^2}{4\pi^2})$

$$A= \frac{C^2}{4\pi}$$

where C is the circumference and A is the area of the circle.


Area of a circle using the circumference – Examples with answers

The following examples are solved using both methods learned to find the area of a circle when we know the circumference. Try to solve the problems yourself before looking at the answer.

EXAMPLE 1

Find the area of a circle that has a circumference of 2 ft.

We can use the first method. Therefore, we are going to find the length of the radius by dividing the circumference by 2π. Thus, we have:

$latex r=\frac{C}{2\pi}$

$latex r=\frac{2}{2\pi}$

$latex r=0.3183$

Now, we use the standard formula for the area of a circle with the value r=0.3183:

$latex A=\pi~r^2$

$latex A=\pi~(0.3183)^2$

$latex A\approx 0.3183$

The area of the circle is $latex 0.31829~{{ft}^2}$.

EXAMPLE 2

If a circle has a circumference of 3 inches, what is its area?

We can use the formula for the area of a circle in terms of the circumference with the value C=3:

$$A= \frac{C^2}{4\pi}$$

$$A= \frac{(3)^2}{4\pi}$$

$$A= \frac{9}{4\pi}$$

$latex A\approx 0.7162$

The area of the given circle is $latex 0.7162~{{in}^2}$.

EXAMPLE 3

Find the area of a circle that has a circumference of 10 feet.

We can start by finding the length of the radius as follows:

$latex r=\frac{C}{2\pi}$

$latex r=\frac{10}{2\pi}$

$latex r=1.5915$

Using the formula for the area of a circle with the value r=1.5915, we have:

$latex A=\pi~r^2$

$latex A=\pi~(1.5915)^2$

$latex A\approx 7.957$

The area of the circle is $latex 7.957~{{ft}^2}$.

EXAMPLE 4

If the area of a circle is 100 feet squared, determine its circumference.

In this case, we already have the area, and we need to determine the circumference. Therefore, we can use the formula for the area of a circle in terms of the circumference and solve for C:

$$A= \frac{C^2}{4\pi}$$

$$100= \frac{C^2}{4\pi}$$

$latex 400\pi={{C}^2}$

$latex C=35.449$

The circumference of the circle is 35.449 feet.

EXAMPLE 5

Find the circumference of a circle that has an area of $latex 88~{{in}^2}$.

We can solve this problem by using the formula for the area of a circle in terms of the circumference and solving for C:

$$A= \frac{C^2}{4\pi}$$

$$88= \frac{C^2}{4\pi}$$

$latex 352\pi={{C}^2}$

$latex C=33.254$

The circumference of the circle is equal to 33.254 in.

EXAMPLE 6

What is the area of a circle that has a circumference of 25 inches?

We are going to use the first method. Therefore, we find the length of the radius as follows:

$latex r=\frac{C}{2\pi}$

$latex r=\frac{25}{2\pi}$

$latex r=3.979$

Now, we use the formula for the area of a circle in terms of the radius:

$latex A=\pi~r^2$

$latex A=\pi~(3.979)^2$

$latex A\approx 49.739$

The area of the circle is 49.739 inches squared.

EXAMPLE 7

Find the area of a circle that has a circumference with a length of 9 yards

We are going to solve this problem using the second method. Therefore, we use the value C=9 in the formula:

$$A= \frac{C^2}{4\pi}$$

$$A= \frac{9^2}{4\pi}$$

$$A= \frac{81}{4\pi}$$

$latex A_{s}\approx 6.446$

The area of the circle is $latex 6.446~{{yd}^2}$.


Area of a circle using the circumference – Practice problems

Use the methods learned to find the area of a circle when we know the circumference. Click “Verify” to check that you got the correct answer.

Find the area of a circle that has a circumference of 4 feet.

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Find the area of a circle that has a circumference of 11 in.

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What is the circumference of a circle that has an area of $latex 30 ~ft^2$?

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If a circle has an area of $latex 60 ~{{ft}^2}$, what is its circumference?

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What is the area of a circle that has a circumference of 20 inches?

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