The diameter of a circle is the line segment that joins two points on the circumference and passes through the center of the circle. **To find the area of a circle using the diameter, we can divide the diameter by 2 to find the radius and use the formula for the area of a circle in terms of the radius.**

In this article, we will learn two methods that we can use to calculate the area of a circle when we know the length of the radius. In addition, we will solve some practice problems.

## How to find the area of a circle using the diameter?

To find the area of a circle when we know the length of the diameter, we can use two main methods.

The first method consists in finding the length of the radius using the diameter and then use it in the formula for the area of a circle. Recall that the diameter can be expressed as follows:

$latex d=2r$

This means that to find the length of the radius, we simply have to divide the length of the diameter by 2.

We can then use the standard Area of a Circle Formula with the radius length 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 diameter.

## Formula for the area of a circle using the diameter

To avoid having to find the length of the radius using the length of the diameter before finding the area of a circle, we can determine a formula for the area of a circle in terms of the diameter.

This way, we can use the diameter length directly in the formula to find the area of a circle in one step.

Therefore, recalling that the formula for a circle is as follows:

$latex A=\pi r^2$

we can substitute the following relationship into the formula:

$latex d=2r$

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

Therefore, we have:

$latex A=\pi r^2$

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

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

where *d* is the diameter of the circle and *A* is its area.

## Area of a circle using the diameter – Examples with answers

Both methods of finding the area of a circle using the diameter are used to solve the following examples. Each example has its respective solution, but try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

Find the area of a circle that has a diameter of 2 feet.

##### Solution

We are going to use the first method. Therefore, we have to find the length of the radius by dividing the radius by 2. Thus, the radius has a length of 1 ft.

Now, we use the standard formula for the area of a circle with the value *r*=1:

$latex A=\pi~r^2$

$latex A=\pi~(1)^2$

$latex A=\pi~(1)$

$latex A=\pi$

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

### EXAMPLE 2

Find the area of a circle that has a diameter of 3 in.

##### Solution

We are going to solve this problem by applying the formula for the area of a circle in terms of diameter

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

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

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

$latex A\approx 7.07$

The area of the given circle is $latex 7.07~{{in}^2}$.

### EXAMPLE 3

If the diameter of a circle is 6 feet, what is its area?

##### Solution

Let’s start by finding the length of the radius. Therefore, we divide the diameter by 2: 6/2=3.

Using the formula for the area of a circle with the value *r*=3, we have:

$latex A=\pi~r^2$

$latex A=\pi~(3)^2$

$latex A=\pi~(9)$

$latex A\approx 28.27$

The area of the circle is $latex 28.27~{{ft}^2}$.

### EXAMPLE 4

If the area of a circle is 100 feet squared, find the length of its diameter.

##### Solution

In this example, we know the area of the circle, and we have to determine the length of the diameter. Therefore, we use the formula for the area in terms of diameter and solve for *d*:

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

$$100=\pi (\frac{d^2}{4})$$

$latex 400=\pi~{{d}^2}$

$latex d^2=127.324$

$latex d=11.28$

Therefore, the diameter of the circle has a length of *d*=11.28 feet.

### EXAMPLE 5

Determine the length of the diameter of a circle that has an area of $latex 55~{{in}^2}$.

##### Solution

This example is similar to the previous one, so we have to use the given area value and solve for *d*:

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

$$55=\pi (\frac{d^2}{4})$$

$latex 220=\pi~{{d}^2}$

$latex d^2=70.028$

$latex d=8.368$

The diameter of the circle has a length of 8.368 in.

### EXAMPLE 6

What is the area of a circle that has a diameter of 12 in?

##### Solution

Using the first method, we are going to find the length of the radius using the diameter. Therefore, we have: 12/2=6.

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

$latex A=\pi~r^2$

$latex A=\pi~(6)^2$

$latex A=\pi~(36)$

$latex A\approx 113.1$

The area of the circle is 113.1 inches squared.

### EXAMPLE 7

Find the area of a circle that has a diameter with a length of 7.5 yards.

##### Solution

We use the value of *d*=7.5 in the formula for the area of a circle in terms of diameter:

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

$$A=\pi (\frac{(7.5)^2}{4})$$

$$A=\pi (\frac{56.25}{4})$$

$latex A_{s}\approx 44.18$

The area of the circle is $latex 44.18~{{yd}^2}$.

## Area of a circle using the diameter – Practice problems

Solve the following problems by applying what you have learned about the area of a circle when we know the length of the diameter. If you have trouble with these problems, you can look at the worked examples above.

## See also

Interested in learning more about circles? You can look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**