What is the Area of a Circle
Area of a circle can be determined easily using a formula – πr2, where r is the radius of the circle. But before knowing more about the area of the circle, let us understand the perimeter of a circle. These two topics are important for circles class 9.
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 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.
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.
Surface Area of a Circle
Take a circle with radius r. Its area, A is given by
A = πr2
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 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.
The area of a rectangle (A) will also be the area of a circle. So, we have
- A = πr×r
- A = 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.
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
If the area of a 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 πr2 /4 square units respectively.
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