Sectors, segments and arcs

In addition to knowing about a circle, it is important to know about various parts of a circle, including arcs, sectors and segments. Do you know what each of these words refers to?

Arcs and sectors are basically just parts of the circle. Arc length is part of the circumference and a sector is part of the area.

The way we calculate both of them is essentially the same. We calculate either the circumference or the area and multiply it by the relevant proportion of the circle.

But how do we find the relevant proportion of the circle? Well, as there are 360 degrees in a circle, the proportion is x/360, where x is the number of degrees that the arc or sector takes up.

Let’s try it with the teacher in these following examples:

Try the exercise below on arc length and sector area (from exercise 5 on pages 113 to 115 of the extended textbook):

Below are the answers:

Segments and chords

Calculating segments is slightly more difficult than calculating sectors, because effectively we treat the segment as the difference between a sector and a triangle, and we normally need to calculate the area of the triangle using trigonometry (specifically the formula Triangle Area = 1/2 (bc)sinA.

So in this diagram, the area of the sector is (⍬/360)x πr² – (1/2)r²sin(⍬).

This can be written as r²((⍬/360)x π – (1/2)sin(⍬))

To find the length of the chord we split the relevant triangle into two right-angled triangles and then use trigonometry (specifically the sine ration) to find the length of their sides that lie along the chord.

Let’s practice some questions on finding the area of segments and the length of chords together:

Try the below exercise to check your knowledge on segments (from exercise 6 on pages 116 and 117 of the extended textbook):

Below are the answers: