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.
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.
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 onebyone

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 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.
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 semicircle and quartercircle with the same radius will be equal to \(\frac{πr^2}{2}\) and \(\frac{πr^2}{4}\) square units respectively.
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