Circle I: Introduction of a Circle & its Properties – SS2 Further Mathematics Lesson Note (Second Term)

A circle is a round shape with a centre and a curved outer boundary called the circumference. Its parts include the arc, radius, chord and diameter, and it can be divided into regions called sectors, semicircles and segments. The length of an arc and the perimeter of a sector are found from the angle at the centre.

Behavioural objectives

By the end of the lesson, students should be able to:

• name and describe the parts of a circle;
• distinguish between a sector, a semicircle and a segment;
• calculate the length of an arc using the angle at the centre;
• calculate the perimeter of a sector.

What are the parts of a circle?

• The centre is the point in the middle of a circle.
• The circumference is the curved outer boundary of the circle.
• An arc is a curved part of the circumference.
• A radius is any straight line joining the centre to the circumference. The plural of radius is radii.
• A chord is any straight line joining two points on the circumference.
• A diameter is a straight line which divides the circle into two equal parts. It is also any chord which goes through the centre of the circle.

What are the regions of a circle?

• A sector is the region between two radii and the circumference.
• A semicircle is the region between a diameter and the circumference, that is, half of the circle.
• A segment is the region between a chord and the circumference.

How do you find the length of an arc and the perimeter of a sector?

Take a circle with centre O and radius r. The circumference of the circle is 2πr.

Length of an arc

The length, L, of arc XY is given as:

L = (θ ÷ 360°) × 2πr

where θ is the angle subtended at the centre by arc XY and r is the radius of the circle.

Perimeter of a sector

The perimeter of sector XOY = r + r + L, where L is the length of arc XY. Therefore:

Perimeter of sector XOY = 2r + (θ ÷ 360°) × 2πr

Worked examples

Example 1

An arc of length 28 cm subtends an angle of 240° at the centre of a circle. In the same circle, what angle does an arc of length 35 cm subtend? (Take π as 22/7.)

Solution

First find the radius, using L = (θ ÷ 360°) × 2πr, with L = 28 cm and θ = 240°.

28 = (240 ÷ 360) × 2 × (22 ÷ 7) × r

28 = (2 ÷ 3) × (44 ÷ 7) × r

28 = 88r ÷ 21

r = (28 × 21) ÷ 88 = 147 ÷ 22 cm

Now use the same radius with L = 35 cm.

35 = (θ ÷ 360) × 2 × (22 ÷ 7) × (147 ÷ 22)

35 = (θ ÷ 360) × 42

θ = (35 × 360) ÷ 42 = 300°

Thus, when the length of the arc is 35 cm, the angle subtended at the centre is 300°.

Example 2

Calculate the perimeter of a sector of a circle of radius 7 cm, the angle of the sector being 108°. (Take π as 22/7.)

Solution

Perimeter of a sector = 2r + (θ ÷ 360°) × 2πr

= 2 × 7 + (108 ÷ 360) × 2 × (22 ÷ 7) × 7

= 14 + (108 ÷ 360) × 44

= 14 + (3 ÷ 10) × 44

= 14 + 132 ÷ 10

= 14 + 13.2

= 27.2 cm

Evaluation

1. Define a chord and a diameter.
2. State the difference between a sector and a segment.
3. Write the formula for the length of an arc and the formula for the perimeter of a sector.
4. Calculate the perimeter of a sector of a circle of radius 14 cm, if the angle of the sector is 90°. (Take π as 22/7.)

Assignment

Calculate the length of an arc and the perimeter of the sector of a circle of radius 21 cm, if the angle of the sector is 60°. (Take π as 22/7.)