Circle II: Length of Chord & Perimeter of a Segment – SS2 Further Mathematics Lesson Note (Second Term)

A chord is a straight line joining two points on a circle. Its length is AB = 2r sin θ, where r is the radius and θ is half the angle the chord subtends at the centre. The perimeter of a segment is the length of the chord plus the length of the arc cut off by the chord.

Behavioural objectives

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

• derive the formula for the length of a chord, AB = 2r sin θ;
• state the formula for the perimeter of a segment;
• find the angle a chord subtends at the centre when the radius and the distance of the chord from the centre are given;
• calculate the length of a minor arc cut off by a chord.

How do you find the length of a chord?

Consider a circle with centre O and radius r. Let OC be the perpendicular distance from O to chord AB, and let angle AOB = 2θ. In the right-angled triangle OCA:

AC/r = sin θ

Cross multiply:

AC = r sin θ

Since AB = 2 × AC,

AB = 2r sin θ

Where:

• r = radius of the circle
• θ = semi-vertical angle of the sector, that is, half of the angle subtended at the centre by arc AB.

What is the perimeter of a segment?

The perimeter of segment ACBD is the length of chord AB plus the length of arc ADB.

Perimeter = 2r sin θ + (2θ/360°) × 2πr

The arc ADB subtends the angle 2θ at the centre, so the arc length uses 2θ.

Worked example

In a circle of radius 6 cm, a chord is drawn 3 cm from the centre.

1. Calculate the angle subtended by the chord at the centre of the circle.
2. Find the length of the minor arc cut off by the chord.

Solution to (a)

Let the required angle AOB = 2θ, where θ is the semi-vertical angle of the sector. Then:

cos θ = 3 cm / 6 cm = 1/2

cos θ = 0.5000

θ = cos^-1 0.5000

θ = 60°

Required angle = 2θ = 2 × 60° = 120°

Solution to (b)

Length of minor arc ADB = (2θ/360°) × 2πr

= (120/360) × 2 × (22/7) × 6 cm

= 4 × 22/7 cm

= 88/7 cm

= 12 4/7 cm

Evaluation

1. Write down the formula for the length of a chord in terms of r and θ.
2. Write down the formula for the perimeter of a segment.
3. A chord of a circle of radius 10 cm subtends an angle of 60° at the centre. Find the length of the chord.
4. A chord is drawn 3 cm from the centre of a circle of radius 6 cm. Find the length of the chord.

Assignment

1. Calculate the area of a sector of a circle of radius 6 cm which subtends an angle of 70° at the centre (π = 22/7).
   A. 44 cm² B. 22 cm² C. 66 cm² D. 11 cm² E. 16.5 cm²
2. What is the angle subtended at the centre of a sector of a circle of radius 2 cm if the area of the sector is 2.2 cm²? (π = 22/7)
   A. 120° B. 31 ½° C. 43° D. 58° E. 63°
3. What is the radius of a sector of a circle which subtends 140° at its centre and has an area of 99 m²?
   A. 18 m B. 27 m C. 9 m D. 30 m E. 24 m
4. A sector of 80° is removed from a circle of radius 12 cm. What area of the circle is left? (π = 22/7)
   A. 253 cm² B. 704 cm² C. 176 cm² D. 125 cm² E. 352 cm²
5. An arc of a circle of radius 7 cm is 14 cm long. What angle does the arc subtend at the centre of the circle?
6. An arc of a circle whose radius is 10 cm subtends an angle of 60° at the centre. Find the length of the arc.