Area Of a Circle

Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula, πr2, where r is the radius of the circle. Apart from the area, circles have a radius, diameter and circumference, as its property. This circumference is similar to the perimeter, where the total distance covered by the boundary of the circle is evaluated.  If the area of a given circle with radius r units is πr2 square units, then an area of semi-circle and quarter-circle with the same radius will be equal to πr2 / 2 and πr/4 square units respectively.

Area of a Circle Formula

Let us take a circle with radius r.

Area of a circle

Then the area for this circle, A, is given by; 

A = πr2

Here, the value of pi, π = 22/7 or 3.14

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

  • Determining the circle’s area using rectangles
  • Determining the circle’s area 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 coloured sectors will contribute to half of the circumference and blue coloured sectors will contribute to 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
  • = πr2

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

A = 1/2×base×height

A = 1/2×(2πr)×r

A = πr2

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 the circle. To define the circumference of the circle, knowledge of a term known as ‘pi’ is required. Consider the circle shown in the fig. 1, with centre at O and radius r.

Circumference of circle

The circumference/Perimeter of the 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, which can be measured by using the formula C = 2πr, where r is the radius of the circle.

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

  • π = Circumference/Diameter
  • π = C/2r
  • C = 2πr

The same is shown in fig. 2.

perimeter of circle

Examples

Question: If radius of a circle is 15cm. Then find its area.

Solution: Given, radius of circle = 15cm

The, area will be;

A = πr2

A = π.152

A = 706.5 sq.cm.

Question: If the diameter of a circle is 10cm. Then find its area.

Solution: Given, diameter = 10cm

So, radius will be = 10/2 = 5cm

Hence, area A = πr2

A = π.52

A= 78.5 sq.cm

Question: If the circumference of a given circle is 30cm. Then what will be its area.

Solution: Given, the circumference of a circle = 30cm

We know, from the formula of circumference, C =2πr

So, we can write,

2πr = 30

or r = 30/2π = 15/π

As we found the value of r, now we can find the area;

A = πr2

A = π(15/π)2

On solving we get,

A = 71.65 sq.cm.

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