Area Of a Circle

Area of a circle can be determined easily using a formula. But before knowing the area of circle, let us understand the perimeter of a circle.

Perimeter of Circle/Circumference of Circle:

A perimeter of closed figures is defined as the length of its boundary. When it comes to circles, the perimeter of a circle is given a different name. It is called Circumference of circle. To define the circumference of circle, knowledge of a term known as ‘pi’ is required. Consider the circle shown in the fig. 1, with center at O and radius r.

A Circle and a Rope

The circumference of the above circle is equal to the length of its boundary. The length of rope which wraps around its boundary perfectly will be equal to its circumference.

π, read as ‘pi’is defined as the ratio of the circumference of a circle to its diameter. This ratio is same for every circle. Consider a circle with radius ‘r’ and circumference ‘C’. For this circle

\(π\) = \(\frac{Circumference}{Diameter}\)

\(π\) = \(\frac{C}{2r}\)

C = 2πr

The same is shown in fig. 2.

Depiction of Length of Boundary of Circle

Area of a Circle

Take a circle with radius r. Its area, A is given by

\(A\) = \(πr^2\)

Area of a circle can be visualized & proved using two methods, namely

  • Determining the area of a circle using rectangles
  • Determining the area of a circle using triangles

Let us understand both the methods one-by-one-

  • Using areas of rectangles

The circle is divided into 16 equal sectors and the sectors are arranged as shown in the fig. 3. The area of the circle will be equal to that of the parallelogram shaped figure formed by the sectors cut out from the circle. Since the sectors have equal area, each sector will have equal arc length. The green colored sectors will contribute for the half of the circumference and blue colored sectors will contribute for the other half. If the number of sectors cut from the circle is increased, the parallelogram will eventually look like a rectangle with length equal to πr and breadth equal to r.

Area of a Circle using Rectangle

The area of a rectangle (A) will also be the area of a circle. So, we have

\(A\) = \(πr~×~r\)

\(A\) = \(πr^2\)

  • Using areas of triangles

Fill the circle with radius r with concentric circles. After cutting the circle along the indicated line in fig. 4 and spreading the lines, the result will be a triangle. The base of the triangle will be equal to the circumference of the circle and its height will be equal to the radius of the circle.

Area of a Circle using Triangles

So, area of the triangle (A) will be equal to the area of the circle. We have

\(A\) = \(\frac{1}{2}~×~base~×~hieght\)

\(A\) = \(\frac{1}{2}~×~(2πr)~×~r\)

\(A\) = \(πr^2\)

If area of a circle with radius r units is \(πr^2\) square units, then area of semi-circle and quarter-circle with the same radius will be equal to \(\frac{πr^2}{2}\) and \(\frac{πr^2}{4}\) square units respectively.

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Practise This Question

Find the area of the shaded region where ABC is a quadrant of radius 5cm and a semicircle is drawn with BC as diameter.