Circles, Arcs and Sectors

The Circle

A circle has three main components:

 

1

 

 

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 .

1

re-arranging this gives the formula

2

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.

 

3

 

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

4

 

5

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.

6

 

Area of a circle

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

7

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.

12

 

ani1

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

Using the equation Area= length x breadth,

 

7

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.

8

 

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.

9 so 10

 

 

Arcs and Sectors - Terminology

  An arc is a part of a curve.
It is a fraction of the circumference of the circle.

arc


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

 

s1

 


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

c1

c2

A chord can be a diameter

c3

 

Arcs and Sectors Equation

11

so

12

 

Arcs  

Example

What is the length of arc AB ?

22

 

13

 

Example

Find the radius of the following circle:

di

 

15

16

 

Sectors

 

 Example

  What is the area  of sector  AOB ?

32

 

14

   
 Example

Find the radius of the following circle:

di2

 

17

18

 

 Example

  What is the length of arc AB ?

42

 

19

 

Chords, Bisectors and Tangents

 

50

 

51

55

 

 

A tangent touches the circle at one point only.

 

52

 

54

 

Pythagoras in a circle

 

     Example

 What is the value of x ?

 

oil

 

                 Move the radius down !

 

oil2

      By Pythagoras’ Theorem

20

 

The angle in a  semi circle

The angle in a semi-circle is 90 °

se

 

ani2

 

 

© Alexander Forrest