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

**Perimeter of Circle/Circumference of Circle:**

Perimeter of closed figures is defined as length of its boundary. When it comes to circles, perimeter of 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.

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 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.

**Area of Circle:**

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

\(A\) = \(πr^2\)

This can be visualized using formula to determine the areas of rectangles and areas of triangles. From here on, these two methods are discussed.

- 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.

The area of rectangle (A) will also be the area of 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.

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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