7 min readConcept sheet1-mark MCQ; 2-mark sector area; part of a case-based question

Area of a sector

A sector is the region between two radii and an arc, like a slice of a round cake. Area is (θ/360) × πr²; the major sector uses 360° − θ.

Area of a sector(θ/360) × πr²

Major sectorπr² − area of the minor sector

Link with the arcArea of a sector = (1/2) × arc length × r

Quadrantθ = 90°, area = πr²/4

Why the formula works

The whole circle is a sector of 360° with area πr². A sector of 1° has area πr²/360. So a sector of angle θ has area (θ/360) × πr². The sector with angle θ less than 180° is the minor sector. The rest of the circle is the major sector, with angle 360° − θ.

How to draw a sector figure

Draw a circle and mark its centre O.
Draw two radii OA and OB.
Mark the angle θ at O with an arc and write its value.
Shade the region between OA, OB and the minor arc AB: this is the minor sector.
Label the unshaded part major sector and write r on one radius.

Minor and major sectors

Problem

A circle has radius 10 cm. Find the areas of its minor and major sectors when the minor sector's angle is 72°. (Use π = 3.14)

Working

Area of the circle = πr² = 3.14 × 10 × 10 = 314 cm².
Minor sector = (72/360) × 314 = (1/5) × 314 = 62.8 cm².
Major sector = 314 − 62.8 = 251.2 cm².

Answer

Minor sector = 62.8 cm², major sector = 251.2 cm²

Word problem: a lawn sprinkler

Problem

A lawn sprinkler sprays water up to 7 m and turns through 120°. Find the area of lawn it waters. (Use π = 22/7)

Working

The watered region is a sector with r = 7 m and θ = 120°.
Area of the full circle = (22/7) × 7 × 7 = 154 m².
Sector = (120/360) × 154 = 154/3.
= 51.33 m² (to 2 decimal places).

Answer

About 51.33 m²

Finding the angle from the area

Problem

A sector of a circle of radius 21 cm has area 231 cm². Find its central angle. (Use π = 22/7)

Working

Area of the circle = (22/7) × 21 × 21 = 1386 cm².
(θ/360) × 1386 = 231.
θ = 231 × 360 ÷ 1386 = 60°.

Answer

60°

In real life

A round cake or pizza cut into 8 equal slices gives sectors of 45° each. A garden sprinkler that turns through an angle waters a sector-shaped patch.

Exam angle

1 marks: “Find the area of a quadrant of a circle of radius 14 cm. (Use π = 22/7)” Quadrant = πr²/4 = (22/7) × 196 ÷ 4 = 154 cm².

2 marks: “Find the area of a sector of a circle of radius 14 cm with central angle 72°. (Use π = 22/7)” πr² = 616 cm². Sector = (72/360) × 616 = (1/5) × 616 = 123.2 cm².

3 marks: “Find the areas of the minor and major sectors of a circle of radius 14 cm, when the minor sector's angle is 135°. (Use π = 22/7)” πr² = 616 cm². Minor = (135/360) × 616 = (3/8) × 616 = 231 cm². Major = 616 − 231 = 385 cm².

Remember it

Sector = the angle's share of the area: θ/360 of πr². The arc goes with 2πr; the sector goes with πr².