The area of a circle is the region occupied by the circle in the two-dimensional plane. We can easily determine this area using the formula **A = πr²**, where

*r*is the length of the radius of the circle and where

*π*is a mathematical constant with an approximate value of 3.14. This formula is useful for finding the space occupied by a circular field or a graph. For example, suppose that if we want to buy a tablecloth for a round table, then this formula allows us to determine the portion of the tablecloth that is necessary to cover it completely.

Here, we will explore the area of a circle in more detail. We will learn how to derive the formula for the area of a circle. In addition, we will solve some exercises where we will apply this formula.

## Formula for the area of a circle

Remember that a circle is a closed geometric figure. Technically, a circle is a set of points located at a fixed distance from a central point. The fixed distance from the point is the radius of the circle. The radius is the line that joins the center of the circle to the outer limit. The following is a circle with radius *r*:

The area of a circle can be calculated with the following formula:

A=πr² |

where, *r* is the length of the radius and *π* is the value of pi, $latex \pi = \frac{22}{7}$ or approximately 3.14.

## Derivation of the formula for the area of a circle

We can derive the formula for the area of circles by dividing the circle into several sectors and arranging the sectors as shown in the following figure:

The area of the circle is equal to the area of the parallelogram formed by the cut sectors of the circle. Since all sectors have the same area, each sector will have the same arc length. 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*.

We know that the area of a rectangle equals base times height, so we have:

$latex A=\pi r\times r$

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

## Finding the area of a circle using the diameter

Recall that the diameter is equal to the distance of the line that crosses through the center of the circle and connects both ends. This distance is equal to twice the radius, that is, we have $latex d = 2r$ or equivalently $latex r = \frac{d}{2}$.

To calculate the area of a circle using the diameter, we have to substitute this equivalence in the area formula:

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

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

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

where *d* is the length of the diameter

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## Area of a circle – Examples with answers

The following examples are solved using the formulas for the area of a circle seen above. Each example has its respective solution, where the process and reasoning used are detailed.

**EXAMPLE 1**

What is the area of a circle that has a radius of 5 m?

##### Solution

We can use the first formula for the area 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 m².

**EXAMPLE 2**

A circle has a radius of 12. What is its area?

##### Solution

Here, we have the radius $latex r=12$, so we use this value in the first formula for the area:

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

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

$latex A=\pi (144)$

$latex A=452.4$

The area of the circle is 452.4 m².

**EXAMPLE 3**

What is the area of a circle that has a diameter of 10 m?

##### Solution

In this case, we have the diameter instead of the radius, so we use the second formula with the value $latex d=10$. Therefore, we have:

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

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

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

$latex A=\pi(25)$

$latex A=78.5$

The area of the circle is 78.5 m².

**EXAMPLE 4**

If a circle has a diameter of 20 m. What is its area?

##### Solution

Again, we use the diameter in the second formula with the value. 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 m².

**EXAMPLE 5**

A circle has an area of 150 cm². What is the length of its radius?

##### Solution

In this case, we start with the area and want to find the length of the radius. Therefore, we use the value $latex A=150$ in the first 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 cm.

## Area of a circle – Practice problems

Use the following problems to practice applying the formulas for the area of circles. Select an answer and verify that it is correct by clicking “Check”. If you need help with this, you can look at the solved examples above.

## See also

Interested in learning more about circles? Take a look at these pages:

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