1. Area of a Circle
A circle is a 2D shape made up of all points that are at the same distance from a fixed point called the centre. This constant distance is called the radius (r).
The area of a circle tells us how much surface is enclosed inside the curved boundary. It is always measured in square units such as \(\text{cm}^2\), \(\text{m}^2\), etc.
1.1. What is Radius?
The radius is the distance from the centre of the circle to any point on the circle. It is the most important measurement because the area of a circle depends entirely on the radius.
If the radius increases, the area increases rapidly, because the radius is squared in the formula.
2. Area Formula
The area of a circle with radius \(r\) is given by one of the most famous formulas in geometry:
\( A = \pi r^2 \)
Here:
- \(A\) = area
- \(r\) = radius
- \(\pi\) is a constant approximately equal to 3.14 or \(\dfrac{22}{7}\)
2.1. Why the Formula Works
The formula comes from breaking the circle into many thin sectors (like slices of a pizza) and rearranging them to form a shape similar to a parallelogram. The height becomes \(r\) and the base becomes \(\dfrac{1}{2} \times \text{circumference} = \pi r\). So the area becomes:
\( A = r \times \pi r = \pi r^2 \)
3. Examples
Let’s look at a few examples to understand how to use the area formula in real problems.
3.1. Example 1
Find the area of a circle with radius \(r = 7\,\text{cm}\).
\( A = \pi r^2 = \pi \times 7^2 = 49\pi \)
Using \(\pi = \dfrac{22}{7}\):
\( A = 49 \times \dfrac{22}{7} = 154\,\text{cm}^2 \)
3.2. Example 2
A circular park has radius \(r = 10\,\text{m}\). The area covered by the park is:
\( A = \pi r^2 = \pi \times 10^2 = 100\pi \)
Using \(\pi = 3.14\):
\( A = 100 \times 3.14 = 314\,\text{m}^2 \)