Perimeter and Area of a Circle – Formulas and Examples

The perimeter of a circle represents the length of the circle’s outline, that is, the perimeter is equal to the circumference. On the other hand, the area is a measure of the region occupied by the circle in the two-dimensional plane. The perimeter of a circle can be calculated using the formula C = 2πr and the area can be calculated using the formula A = πr², where r is the radius of the circle.

In this article, we will learn about the perimeter and area of a circle in detail. We will learn about its formulas and apply them to solve some practice problems.

GEOMETRY
Formulas for the perimeter and area of a circle

Relevant for

Learning about the perimeter and area of a circle with examples.

See examples

GEOMETRY
Formulas for the perimeter and area of a circle

Relevant for

Learning about the perimeter and area of a circle with examples.

See examples

How to find the perimeter of a circle?

We can calculate the perimeter of a circle, also known as the circumference, by multiplying the length of its diameter by the constant pi. Therefore, we have the following formula:

$latex C=\pi d$

Here, d is the length of the diameter and π is a mathematical constant with a value of approximately 3.141592…

To understand the definition of π, we can consider the following circle:

diagram of the perimeter of a circle

As we already mentioned, the perimeter is equal to the circumference, since it is the length of the outline of the circle. Also, we see that the diameter is the distance that crosses the circle and passes through the center. The constant π is the ratio of the circumference to the diameter.

Therefore, the definition of π is as follows:

$latex \frac{C}{d}=\pi$

Rearranging the formula, we have the formula for the circumference or perimeter of the circle:

$latex C=\pi d$

Calculate the perimeter of a circle with the radius

To calculate the perimeter of a circle using the radius, we recall that the diameter is equal to 2r. Therefore, we can rewrite the perimeter formula as follows:

$latex C=2\pi r$

How to find the area of a circle?

To calculate the area of a circle, we can multiply π by the radius of the circle squared. Therefore, we have the following formula:

$latex A=πr^2$

Remember that the radius of a circle is the segment that joins the center of the circle with the circumference. The following is a circle with radius r:

diagram of a circle with radius

Proof of the formula for the area of a circle

The formula for the area of a circle can be proved using various techniques. Some of the more advanced techniques include the use of Calculus. However, here we are going to simplify the demonstration a bit by using the following diagram:

derivation of the formula for the area of a circle

If we cut the circle into several sectors, we can rearrange the sectors and form a parallelogram-shaped figure. If we make small enough cuts, we will eventually form a parallelogram.

If the number of sectors cut from the circle is increased, the parallelogram will eventually look like a rectangle with a base equal to πr and a height equal to r.

Since the area of a rectangle is equal to the length of its base multiplied by its height, we have:

$latex A=\pi r\times r$

$latex A=\pi {{r}^2}$

Calculate the area of a circle using the diameter

To calculate the area of a circle using the diameter, we remember that the diameter is equal to 2r. Equivalently, we can write $latex r=\frac{d}{2}$. Therefore, we use this equivalence in the formula for the area of a circle:

$latex A=\pi {{r}^2}$

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

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

where, d is the length of the diameter.


Perimeter and area of a circle – Examples with answers

The formulas for the perimeter and area of a circle are used to solve the following examples. Try to solve the problems yourself before looking at the solution.

EXAMPLE 1

Find the perimeter of a circle that has a diameter of 6 feet.

We use the value $latex d=6$ in the formula for the perimeter in terms of the diameter:

$latex C=\pi d$

$latex C=\pi (6)$

$latex C=18.8$

The perimeter of the circle is equal to 18.8 ft.

EXAMPLE 2

If a circle has a radius of 5 inches, what is its area?

We are going to use the formula for the area in terms of the radius with the value $latex r=5$. Therefore, we have:

$latex A=\pi {{r}^2}$

$latex A=\pi {{(5)}^2}$

$latex A=\pi (25)$

$latex A=78.5$

The area of the circle is 78.5 in².

EXAMPLE 3

Find the perimeter of a circle that has a diameter of 15 in.

We apply the formula for the perimeter of a circle with the value $latex d=15$:

$latex C=\pi d$

$latex C=\pi (15)$

$latex C=47.1$

The perimeter is equal to 47.1 in.

EXAMPLE 4

Find the area of a circle that has a radius of 12 inches.

Using the formula for the area of a circle with the value $latex r=12$, we have:

$latex A=\pi {{r}^2}$

$latex A=\pi {{(12)}^2}$

$latex A=\pi (144)$

$latex A=452.4$

The area of the circle is 452.4 in².

EXAMPLE 5

Find the length of the perimeter of a circle that has a radius of 8 yd.

Here, we have the length of the radius instead of the diameter, so we use the value $latex r=8$ in the following formula:

$latex C=2\pi r$

$latex C=2\pi (8)$

$latex C=50.3$

The perimeter is equal to 50.3 yd.

EXAMPLE 6

Find the area of a circle that has a diameter of 10 ft.

We use the formula for the area of a circle in terms of the diameter with the value $latex d=10$:

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

$latex A=\pi(\frac{{{(10)}^2}}{4})$

$latex A=\pi(\frac{100}{4})$

$latex A=\pi(25)$

$latex A=78.5$

The area of the circle is 78.5 ft².

EXAMPLE 7

Find the perimeter of a circle that has a radius of 13 in.

We use the value $latex r=13$ in the formula for the perimeter of a circle. Therefore, we have:

$latex C=2\pi r$

$latex C=2\pi (13)$

$latex C=81.7$

The perimeter is equal to 81.7 in.

EXAMPLE 8

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

We are going to use the formula for the area of a circle in terms of the diameter. Therefore, we substitute the value $latex d=20$:

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

$latex A=\pi(\frac{{{(20)}^2}}{4})$

$latex A=\pi(\frac{400}{4})$

$latex A=\pi(100)$

$latex A=314.16$

The area of the circle is 314.16 in².

EXAMPLE 9

What is the diameter of a circle that has a perimeter of 100 feet?

In this example, we know the perimeter of the circle, and we are going to find the diameter, so we use the value $latex C=100$ in the formula and solve for d:

$latex C=\pi d$

$latex 100=\pi d$

$latex d=\frac{100}{\pi}$

$latex d=31.8$

The length of the diameter is equal to 31.8 ft.

EXAMPLE 10

Find the length of the radius of a circle that has an area of 150 in².

In this example, we know the area, and we are going to find the length of the radius. Therefore, we use the value $latex A=150$ in the area formula and solve for r:

$latex A=\pi {{r}^2}$

$latex 150=\pi {{r}^2}$

$latex {{r}^2}=\frac{150}{\pi}$

$latex {{r}^2}=47.75$

$latex r=6.91$

The radius of the circle is 6.91 in.


Perimeter and area of a circle – Practice problems

Solve the following problems using what you have learned about the perimeter and area of a circle. Click “Check” to make sure that you got the correct answer.

Find the length of the perimeter of a circle that has a diameter of 11 feet.

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What is the area of a circle that has a radius of 15 feet?

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Determine the length of the perimeter of a circle that has a radius of 9 inches.

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Find the area of a circle that has a diameter of 16 inches.

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Find the length of the diameter of a circle that has a perimeter of 50 inches.

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If a circle has an area of 254.5 ~$latex{{yd}^2}$, what is its radius?

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