# Circles, Arcs and Sectors

A circle has three main components:

- The circumference, which is the outside edge of the circle.
- The radius, which goes from any point on the circumference to the centre of the circle.
- The diameter, which joins the circumference through the centre of the circle.

## Calculating the circumference

A relationship exists between all circles, such that the circumference divided by the diameter always has the same ratio.

This ratio is called Pi, the 16th letter of the Greek alphabet,which is an irrational number and has the symbol π .

re-arranging this gives the formula

Very often, π is taken to 2 decimal places and uses the value 3.14

Example

Find the circumference of a circle which has a diameter of 4 cm.

Use the pi button on your calculator and give your answer correct to two decimal places.

Since the diameter of a circle is twice its radius, d=2r

Example

Find the circumference of a circle which has a radius of 4 cm.

Use the pi button on your calculator and give your answer correct to two decimal places.

To find the area of a circle, you could attempt to count the number of squares inside it.

This would give an approximation of the area.

You could also cut the circle into segments, and lay them out next to each other.

Here, the circle is cut into 8 equal parts.

As the circle is cut into smaller and smaller parts, a rectangle is formed.

Using the equation Area= length x breadth,

Example

Find the area of a circle which has a radius of 4 cm.

Use the pi button on your calculator and give your answer correct to two decimal places.

Example

Find the area of a circle which has a diameter of 4 cm.

Use the pi button on your calculator and give your answer correct to two decimal places.

so

## Arcs and Sectors - Terminology

An arc is a part of a curve.

It is a fraction of the circumference of the circle.

A sector is part of a circle enclosed between two radii.

A chord is a line joining two points on a curve.

A chord can be a diameter

## Arcs and Sectors Equation

so

Example

What is the length of arc AB ?

Example

Find the radius of the following circle:

Example

What is the area of sector AOB ?

Example

Find the radius of the following circle:

Example

What is the length of arc AB ?

## Chords, Bisectors and Tangents

A tangent touches the circle at one point only.

Example

What is the value of x ?

Move the radius down !

By Pythagoras’ Theorem

## The angle in a semi circle

The angle in a semi-circle is 90 °

© Alexander Forrest