The area of a circle is the 2D space within the circle's boundary. Knowing how this area relates to the other features of the circle can let us calculate the area of a circle from its other features or it can be used to find different measurements of the circle with a given area.
We have defined the area of a unit square, then used this definition to find the areas of rectangles, triangles, and various other polygons. This works because polygons have straight edges, and we can divide them up and rearrange them into other shapes for which we already know the area.
Circles don't have straight edges, but we can still do something similar to find an expression for the area of a circle - by using a set of isosceles triangles which share a common vertex at the centre of the circle to approximate the area of the circle.
Take a look at the following applet by moving the sliders:
The more triangles we use, the closer this area gets to the area of the circle, and the closer the base of the resulting parallelogram gets to being half of the circumference of the circle. So we can see that the area of a circle is given by \pi r^2.
We can calculate the area of a circle using the formula \text{A}=\pi r^2 where A is the area and r is the radius of the circle.
Using this formula, we can find the area of a circle using its radius and vice versa.
Consider the circle below:
What is the exact area of the circle?
What is the area of the circle rounded to two decimal places?
A circle has an area of 16\pi \text{ cm}^2.
What is its exact radius?
We can calculate the area of a circle using the formula:
This formula can also be used to find the radius if we know the area of the circle.