The Area of Circle formula is:
But how does that come about?
This page looks to give a general run through of how the formula for the area of a circle can be derived.
First, consider 2 regular Polygons inside a standard circle.
A regular Polygon, sometimes referred to as n-gon, is a shape with straight edges that are all the same length.
As can be seen, the 8 sided Polygon on the right, is closer to having the shape of a circle than the 6 sided Polygon on the left.
In fact, the more and more sides a regular Polygon has, the closer and closer the overall shape gets to actually being a circle itself.
Now using the 6 sided Polygon inside a circle as our point of reference.
This Polygon can be separated into a number of isosceles triangles, as can any regular Polygon inside a circle.
An isosceles triangle is a triangle where 2 of the 3 sides are of equal length.
The number of triangles depends on the number of sides. A 6 sided Polygon can have 6 triangles, an 8 sided Polygon 8 triangles, etc.
So the more sides a Polygon has, the more triangles that can be made.
The important part to note, is that the 2 longer sides of the triangle are the same length as the radius.
The area of a triangle is given by   × base × height,  ( bh ).
In each Polygon of n sides, there are n triangles.
So the area of a Polygon of n sides inside a circle is:
bh x n.
Through the properties of multiplication, this can be rewritten as:
The value of bn, is the length of the base of 1 triangle, multiplied by n sides.
Thus the value bn is the perimeter of an n sided Polygon.
When n is very large, and there are lots and lots of sides in the Polygon inside a circle.
The perimeter given by bn will be very close to the actual circumference of the actual circle. Like with how an 8 sided Polygon was closer to a circle than a 6 sided Polygon.
The higher number of sides n, the closer the shape of the Polygon is to a circle.
Knowing this, bn can be replaced by the formula for the circumference of a circle, 2πr.
Giving   × 2πr.
Also, when the number of sides n is very large, the length of b in each triangle will get smaller and smaller, approaching 0 in fact.
The more sides in the Polygon, the smaller the length of b.
As the length of b gets smaller approaching zero, the height of each triangle h, will get closer to becoming the same length as the circle radius r.
Knowing this, h can be replaced by r.
Giving   × 2πr.
Now from here:
× 2πr   =     =   πr2