This page describes how to derive the formula for the area of a circle. we start with a regular polygon and show that as the number of sides gets very large, the figure becomes a circle. By finding the area of the polygon we derive the equation for the area of a circle.

Try this.
In the applet below we have a six-sided regular polygon.
Keep clicking on 'more' and note that as the number of sides gets larger, the polygon approaches being a circle.
As the number of sides becomes infinitely large, it is, in fact, a circle. Click 'reset' afterwards.

The polygon can be broken down into *n* isosceles triangles (where *n* is the number of sides),
such as the one shown on the right.

In this triangle

*s* is the side length of the polygon

*r* is the radius of the polygon and the circle

*h* is the height of the triangle.

The area of the triangle is half the base times height or
There are *n* triangles in the polygon so
This can be rearranged to be
The term *ns* is the perimeter of the polygon (length of a side, times the number of sides).
As the polygon gets to look more and more like a circle, this value approaches the circle circumference, which is *2πr*.
So, substituting *2πr* for *ns*:
Also, as the number of sides increases, the triangle gets narrower and narrower, and so when *s* approaches zero, *h* and *r* become the same length.
So substituting *r* for *h*:
Rearranging this, we get

The radius *r* of a circle is half the diameter *d*
Substituting *r* into the area formula
Which simplifies to

The circumference *c* of a circle radius *r* is given by
Dividing both sides by 2π
Substitute this into the area formula for *r*
Which simplifies to

- Circle definition
- Radius of a circle
- Diameter of a circle
- Circumference of a circle
- Parts of a circle (diagram)
- Semicircle definition
- Tangent
- Secant
- Chord
- Intersecting chords theorem
- Intersecting secant lengths theorem
- Intersecting secant angles theorem
- Area of a circle
- Concentric circles
- Annulus
- Area of an annulus
- Sector of a circle
- Area of a circle sector
- Segment of a circle
- Area of a circle segment (given central angle)
- Area of a circle segment (given segment height)

- Basic Equation of a Circle (Center at origin)
- General Equation of a Circle (Center anywhere)
- Parametric Equation of a Circle

- Arc
- Arc length
- Arc angle measure
- Adjacent arcs
- Major/minor arcs
- Intercepted Arc
- Sector of a circle
- Radius of an arc or segment, given height/width
- Sagitta - height of an arc or segment

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