A circle is a locus of the points equidistant from a common point called the centre. It is a two-dimensional shape that has area and perimeter. Area of a circle is the region enclosed by it in a 2D plane. The perimeter of the circle also termed as the circumference is the measure of the boundary of the circle. 

In geometry or mathematics, a circle can be defined as a special variety of ellipse in which the eccentricity is zero and the two foci are coincident.

What is a Circle: Parts and Properties

Let us look back at a circle, its parts and properties before we learn about area of circle in detail. A circle is a round shape made up of all the points on a flat surface that are the same distance away from a special point called the centre. You can think of it as the path created by a point moving on the surface, but always staying the same distance from the centre point. 

It is a basic 2D shape measured in terms of the length of its radius. It divides the 2D plane into two regions: interior and exterior. 

Parts of a Circle

Some of the parts of a circle are mentioned below:

Radius: Radius of a circle is defined as the distance of a point on the boundary of the circle from the center of the circle. We represent the radius of the circle as ‘r’ or ‘R’.
Diameter: Diameter of a circle is the line whose endpoints lie on the boundary of the circle and pass through the center of the circle. We represent the diameter of the circle as ‘d’ or ‘D’. Also, the diameter of the circle is double in measure than the radius of the circle.
Circumference: Circumference of a circle is the length of the boundary of the circle. So, we can say that the perimeter of the circle is equal to the circumference of the circle.

Properties of a circle

A circle has following properties:

• A circle has the biggest area compared to any other shape with the same distance around it.
• A circle has lots of symmetry. If you draw a line through the middle, it looks the same on both sides. You can also turn it around and it still looks the same.
• All circles are similar, which means they have the same shape but can be different sizes.
• The area inside a circle and the square of its radius are also in the same ratio. This ratio is the famous number π.
• The unit circle is a special circle with a radius of 1 unit and area 1 sqaure unit.

What is Area of a Circle?

Area of a Circle is the region involved by the circle in a two-dimensional plane. It is determined by using the formula: Area of Circle = \(\pi r^2\), where r is the radius of the circle.

The unit of area of a circle is the square unit such as \(m^2,cm^2,\ and\ so\ on.\)

Area of a Circle Formulas

The area is the measure of space confined within the boundary of a circle. The field within the boundary of the circle is the area utilized by the circle. It may also be regarded as the total number of square units inside that circle.

Area of a Circle: \(=\pi r^{2\ }\ or\ \frac{\pi d^2}{4}\ \text{in square units}\) \(where\ \pi=\frac{22}{7}or\ 3.14 \text{ where d is the diameter of the circle}\)

Example of area of a circle:

\(\text{Find the circumference and the area of a circle whose radius is 21 cm.}\)
\(\text{Given}:\)
\(\text{Radius of the circle} =21\ cm.\)
\(\text{Circumference of the Circle}=\ 2\pi r\ =2\times\frac{22}{7}\times21\)
\(=2\times22\times3=132\ cm\)
\(\text{Using the area of Circle formula}=\pi r^2\)
\(=\frac{22}{7}\times21\times21=1386\ sq.cm.\)
\(\text{Area of the Circle}=1386\ sq.cm\ or\ 1386\ cm^2\)

How to Find Area of a Circle?

We know that the area of a circle can be written as: \(\pi \times r^2\).
Here ‘r’ is the radius of the circle.

In order to find the area of the circle we need to know either the radius of the circle or its diameter. Also, we can find the area of the circle using the circumference of the circle. Let us learn step by step how can we find the area of a circle when either the radius or the diameter is given and also when the circumference of the circle is given:

Area of a Circle using Diameter

We already know that the diameter of a circle is double its radius.
If the radius of the circle is ‘r’, then diameter is ‘2r’
We can write d = 2r or r = d/2

So, the formula for area of a circle using diameter becomes = \(\frac{\pi }{4}\times d^2\)
Let us understand this using an example:

Example: Find the area of a circle whose diameter is 4cm.

Solution:
Step 1: First we need to note down the diameter of the circle. In this case, d = 4cm
Step 2: Write the formula for the area of the circle using diameter.
Area = \(\frac{\pi }{4}\times d^2\)

Step 3: Replace ‘d’ with the value of ‘d’ in the above formula, i.e. 4 cm.
Area = \(\frac{\pi }{4}\times 4^2\)

Step 4: Simplify the equation to get the answer in square units.

Area = \(\frac{22}{7\times 4}\times 4\times 4\) sq. cm
Area = \(\frac{22}{7}\times 4\)sq. cm
Area = \(\frac{88}{7}\)sq. cm
Area = 12.57 sq. cm.

Area of a Circle using Circumference

We know that the circumference of the circle is the length of the boundary of the circle. It is also termed as the perimeter of the circle.
Also, we know that the circumference of a circle is determined by the formula = \(2\pi r\)
Here, ‘r’ is the radius of the circle.

Now, to calculate the area of the circle using circumference, we use the formula:
Area = \(\frac{C^2}{4\pi }\), here C is the circumference of the circle.

Example: Find the area of a circle whose circumference is 80 cm.

Solution:
Step 1: First we need to note down the circumference of the circle. In this case, C = 80cm.
Step 2: Write the formula for the area of the circle using circumference.
Area = \(\frac{C^2}{4\pi }\)
Step 3: Replace ‘C’ with the value of ‘C’ in the above formula, i.e. 80 cm.
Area = \(\frac{80^2}{4\pi }\)
Step 4: Simplify the equation to get the answer in square units.

Area = \(\frac{80\times 80\times 7}{4\times 22}\)sq. cm
Area = \(\frac{20\times 40\times 7}{11}\)sq. cm
Area = \(\frac{800\times 7}{11}\)sq. cm
Area = \(\frac{5600}{11}\)sq. cm
Area = 509.09 sq. cm

Derivation of Area of a Circle

The area of a circle can be determined using the below 2 methods:

1. Determining the circle’s area using rectangles
2. Determining the circle’s area using triangles

Derivation of Circle’s Area using Area of Rectangles

Recognize the below figure, if we divide up the circle into smaller sections and organize them in a systematic manner. It develops a shape of a parallelogram. When the circle is split into even smaller sectors, it progressively takes the shape of a rectangle. The more the number of sections the more it tends to have a shape of a rectangle as shown below.

Area of a rectangle from the above figure is \(A=l\times b=\pi r\times r=A=\pi r^2\) which is equal to the area of the circle.

Derivation of Circle’s Area using Areas of Triangles

Cover the circle with concentric circles of “r” radius. After dividing the circle along the designated line shown in the above figure and spreading the lines, the outcome will be a triangle. The base of the triangle will be equivalent to the circumference of the circle, and its height will be identical to the radius of the circle. Below is the formula calculation for the same.

Area of the triangle from the above figure is \(A=\frac{1}{2}\times base\times height=A=\frac{1}{2}\times(2\pi r)\times r=A=\pi r^2\) which is equal to the area of the circle.

Area of a Circle from Coordinates

The area of a circle can be calculated using the formula \(A = \pi r^2\), where \(A\) represents the area and \(r\) denotes the radius of the circle.

In some cases, we may not have the radius explicitly given, but instead, we have the coordinates of the circle's centre and a point lying on the circumference.

To find the area of such a circle, we can follow these steps:

• Use the distance formula to determine the radius by calculating the distance between the centre point \((x_1, y_1)\) and the point on the circumference \((x_2, y_2)\).
• Apply the distance formula: \(r = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\).
• Once the radius is determined, substitute it into the formula \(A = \pi r^2\) to obtain the area.

Example:

Consider a circle with the centre at \((3, 4)\) and a point on the circumference at \((6, 8)\).

Using the distance formula:

Calculate the radius: \(r = \sqrt{{(6 - 3)^2 + (8 - 4)^2}} = \sqrt{18}\).

Substituting the radius into the area formula:

Determine the area: \(A = \pi (\sqrt{18})^2 = 18\pi\).

By following these steps, we can find the area of a circle when given the coordinates of its centre and a point on the circumference.

Area of a Sector of a Circle

The area of a sector of a circle is the portion of the area taken within the boundary of a sector of a given circle. Now how to estimate the area of a sector. For this if the radius (r) and the angle of the sector (that is θ) are given to us, then:

Area of a circle=\(π \times r^{2}\)

Area of sector (A) =\(\left(\frac{θ}{360^\circ}\right)\times\pi r^2\)

Solved Examples of Area of a Circle

Having read about all the formulas, along with the definition and derivation using rectangles and triangles. Let us practise some examples for more clarity:

Example 1: Find the circumference and the area of a circle whose radius is 21 cm.

Solution: Given

The radius of the circle =21 cm.

Circumference of the Circle=

\(2\pi r =2\times\frac{22}{7}\times21\)

\(=2\times22\times3=132\ cm\)

Using the area of Circle formula=\(\pi r^2\)

\(=\frac{22}{7}\times21\times21=1386\ sq.cm.\)

Area of the Circle= \(1386\ sq.cm\ or\ 1386\ cm^2\)

Example 2: The proportion of the area of two circles is 9:16. Applying the area of the circle concept determines the ratio of their radii.

Solution: Consider the below assumptions:

The radius of the circle 1= \(r_1\) and area = \(A_1\)

Similarly if the radius of the circle 2 = \(r_2\) and area = \(A_2\)

We are given that \(A_1:A_2 = 9:16\)

Area of a Circle =\(\pi r^2\)

\(πr_1^2 : πr_2^2 = 9:16\)

Hence we get:

\(r_1^2 : r_2^2 = 9:16\)

\(r_1 : r_2=3:4\)

Hence, the ratio of the radii = 3:4

Example 3: Obtain the radius of the circle whose area is 88 sq.cm.

Solution: By the formula:

A = \(\pi r^2\)

Now, substituting the value:

\(\pi r^2=88\)

\(\frac{22}{7}\times r^2=88\)

\(r^2=88\times\frac{7}{22}\)

\(r=\sqrt{4\times7}=5.29\)

Example 4: Find the area of the circle, if we are given that the longest chord of a circle is 28 cm.

Solution: It is known to us that the longest chord of a circle is the diameter.

Therefore, d = 28 cm.

So, r = d/2 = 28/2 = 14 cm.

As per the formula:

A = \(\pi r^2\) square units.

Substituting r = 14 cm in the above formula, we obtain;

\(A=\pi r^2\)

\(A=\frac{22}{7}\times r^2\)

\(A=\frac{22}{7}\times14\times14\)

\(A=616\ cm^2\)

Example 5: For building a house we need to drill holes and fill them with concrete. These holes are 0.4 m wide and 1m deep. Find the amount of concrete required to fill the hole.

Solution: As the holes are circular in shape, so, first we need to find the area of the holes.

Given that:
Diameter of the hole = 0.4 m
Therefore, Area = \(\frac{\pi }{4}\times d^2\)
Let us replace ‘d’ with 0.4 m in the above formula:

Area = \(\frac{\pi }{4}\times \left(0.4\right)^2\)sq.m
Area = \(\frac{22}{7\times 4}\times \frac{4}{10}\times \frac{4}{10}\)sq.m
Area = \(\frac{22}{7}\times \frac{1}{10}\times \frac{4}{10}\)sq.m
Area = \(\frac{88}{7}\times \frac{1}{100}\)sq.m
Area = \(12.57\times \frac{1}{100}\)sq.m
Area = 0.126 sq.m

Also, as the hole is 1m deep:

Volume of the hole or amount of concrete required to fill the hole = \(Area \times Depth\)
Volume = \( 0.126 sq. m \times 1 m\) = 0.126 cu. m.

Therefore, the amount of concrete required to fill the hole is 0.126 cu. m.

We hope that the above article on Area is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.