Area of a Circle

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What is a Circle?

A circle can be defined as a closed plane geometric shape. When it comes to technical terms, a circle can be defined as a locus of a point moving around a fixed point. This is done at a fixed distance away from the central point. This means that the area of the circle is equal to the area that is enclosed within a closed curve with its outer line equidistant from the center. Also, the fixed distance from the center point is known as the radius of the circle.


It should be noted that a radius of a circle can be represented by ‘r’ or ‘R.’ One should remember this because radius plays an important role when it comes to the area of a circle equation.


In this article, students will be able to learn about the area of circle formula, area and perimeter of circle formula, derivation of area of circle, and how the area of a circle definition is used to arrive at other formulas.


Another important concept that is related to the formula to find the area of a circle is the diameter. The diameter of a circle is the line that divides the entire circle into two equal parts. It can also be said that the double of the radius of a circle is known as the diameter. It is represented by ‘d’ or ‘D.’ Further, the radius and diameter of a circle are related to one another by the following equation.

d = 2r or D = 2R

Also, if the value of the diameter of a circle is given, then we can also find the value of the radius of the circle by using the formula that is mentioned below.

r = d / 2 or R = D / 2

Both of these formulas come in handy when students have to solve questions related to what is the area of a circle. Further, the area of a circle can be expressed in several units. In most questions, the units are centimeters or meters.


Area of a Circle

In this section, we will answer the question of what if the formula for the area of a circle. But we will begin with a theoretical groundwork. All students must know that any geometrical shape has its own area. The area can be defined as the region that is occupied by the shape in a two-dimensional plane.


This means that the area of a circle is the total area that is covered by one complete cycle of the radius of the circle on a two-dimensional plane. But how does one calculate this area? The answer is simple. All you need is the formula of the area of a circle.


Let’s derive this formula. Assume that there is a circle with radius r. The circle has a center O and the area, A, of the circle is equal to the product of pi and the square of the radius. This means that:

Area of the circle, A = π r2 square units

Here, in the formula, the value of pi, π = 22 / 7 or 3.14 and r is the radius.

You can see a visual representation of this with the help of the image that is attached below.


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Now, we know about the area of a circle. So, let’s try to learn the surface area of a circle. We know that a circle is nothing more than a representation of a sphere on a two-dimensional plane. This means that the total area that is considered inside the boundary of the circle is the surface area of the circle.


This is why when we say that we want to find the area of a circle, then we refer to the surface area of the circle in fact. In some cases, the volume of a circle can also help in defining the area of a circle.


It is also possible for students to find out the surface area of a circle if the length of the radius, diameter, or even the circumference of the circle is mentioned. The formula for calculating the surface area of a circle is = A = π x r2.


The next concept that students need to learn about a circle is the circumference of a circle. Students must be familiar with the fact that a perimeter of closed figures can be defined as the length of the boundary. Similarly, when it comes to circles, then the perimeter is just given with a different name.


In the case of a circle, the perimeter is known as the circumference of the circle. The circumference is the length of the boundary of the circle. This refers to the fact that if we were to open a circle to form a straight line, then the length of that straight line will be equal to the value of the circumference.


If a student wants to find the value of the circumference, then he or she must have knowledge about pi or π. Now, if you are familiar with the concept of pi, then, let’s proceed further. If we have a circle with a center O and a radius r, then the perimeter of the circle should be equal to its boundary.


Hence, the formula for calculating the circumference is:

Circumference or perimeter of a circle = 2 π r units

If you want a visual representation of this formula, then you should refer to the image that is mentioned below.


(Image will be uploaded soon)


In this formula, r is the value of the radius. For students who are unfamiliar with the concept of pi, it is the ratio of the circumference of a circle to its diameter. The ratio is the same for every circle. This means that if there is a circle with radius ‘r’ and circumference ‘c,’ then:

  • π = circumference / diameter

  • π = c / 2r

  • C = 2 π r

A visual representation of all these formulas and conclusions is given below.


(Image will be uploaded soon)


But how can one arrive at the area of a circle if he or she only knows about the value of circumference of the same circle? In that case, the student should use the formula that is mentioned below.

C = 2 π r

We can also use the same formula to find out the value of the radius. The formula for finding out the radius will be:

R = C / 2 π

It is possible for students to find the value of the square area and circle area. This is because we are familiar with the fact that the area of a circle is estimated to be 80% of the area of a square. This is when the diameter of a circle and the length of the side of the square is the same.


It might also be interesting for students to do an activity by inserting a circular object into a square shape with the same diameter and side length, respectively. If the area of the square is 100 square units, then the area of the circle will be around 80 square units of it.


Fun Facts About How to Find the Area of a Circle

Did you know that the area of a circle has two interesting derivations? The area of the circle can be used for determining the area of a rectangle and the area of a triangle! Let’s first discuss the derivation of the area of a rectangle from a circle.


According to experts, the circle can be divided into 16 equal sectors. All of these sectors can be arranged in a particular form that is depicted in the image that is attached below.


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From this image, you might see that the area of a circle will be equal to that of the parallelogram-shaped figure on the side constructed from the sectors that were cut out of the circle. We also know that the sectors have equal areas. This further means that every sector will have an equal arc length.


The red-coloured sectors will contribute towards half of the circumference and the blue colored sectors will contribute to the other half. This means that if the number of sectors that are cut from the circle is increased, then the parallelogram will start to look like a rectangle. This rectangle will have a length that is equal to π r and breadth that is equal to r. Hence, the area of the rectangle, A, will also be similar to the area of a circle. This means that the formula for A is:

A = π r x r

A = π r2

We can also derive the value of the area of a triangle in a similar fashion. If we fill the circle with radius r with concentric circles and cut the circle along the indicated lines, then we will get a triangle after spreading the lines. A visual representation of this is available in the image that is mentioned below.


(Image will be uploaded soon)


From this image, we can see that the area of a triangle (A) will be equal to the value of the area of a circle. This means that:

A = ½ x base x height

A = ½ x (2 π r) x r

A = π r2

FAQ (Frequently Asked Questions)

Question 1. How to Find the Area of a Circle Whose Diameter is Given?

Answer: The area of a circle whose diameter is given can be calculated in two methods. The shorter method is to use the formula of A= π(d/2)2. Also, you can calculate the radius of the circle by dividing the given diameter by 2 and then use the formula A= π r2.

Question 2. Mention the Different Formulas that one can Use to Calculate the Area of a Circle.

Answer: The formulas that can be used for calculating the area of a circle are:

Area = π x r2 (This is in terms of the radius ‘r’).

Area = (π / 4) x d2 (This is in terms of the diameter ‘d’).

Area = C2 / 4 π (This is in terms of the circumference ‘c’).