Random articles

Short Multiplication Prime Numbers Simplifying Expressions with Exponents Palindrome Numbers, Compatible Numbers Polyhedrons Perimeter of a Segment Ordering Mixed Fractions & Negative FractionsThe 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 \bf{\frac{1}{2}} ×

As we've seen, in each Polygon of

So the area of a Polygon of

\bf{\frac{1}{2}}

Through the properties of multiplication, this can be rewritten as:

\boldsymbol{\frac{h}{2}}

The value of

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 \boldsymbol{\frac{h}{2}} **×**
**2πr**.

When the number of sides

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 \boldsymbol{\frac{r}{2}}

Now from here:

\boldsymbol{\frac{r}{2}}

- Home ›
- The Circle › Area of Circle Formula