The rest of the circle
Take the minor piece away from the whole circle, and the rest is the major piece.
A slice is taken away
A round pizza of radius 4 cm has area 50.24 cm² (π = 3.14). A 30° slice, of area 4.19 cm², is taken away.
What is the area of the pizza that is left?
What this lesson covers
The idea
Area of the major sector = πr² – area of the minor sector, which equals (360 – θ)/360 × πr²; area of the major segment = πr² – area of the minor segment.
A slice is taken away
A round pizza of radius 4 cm has area 50.24 cm² (π = 3.14). A 30° slice, of area 4.19 cm², is taken away.
What is the area of the pizza that is left?
- 4.19 cm²
- 46.05 cm²
- 54.43 cm²
Find the major piece
The Sector tab shows a circle of radius 4 cm. The Segment tab shows a circle of radius 14 cm. Change the angle and read the area of the major piece in the working.
Whole circle minus the minor piece
Area of the major sector = πr² − area of the minor sector = (360 − θ)/360 × πr². Area of the major segment = πr² − area of the minor segment.
For r = 4 cm and θ = 30°: 50.24 − 4.19 = 46.05 cm². Or directly, 330/360 × 3.14 × 16 = 46.05 cm².
For r = 14 cm and θ = 90°: the circle is 616 cm² and the minor segment is 56 cm², so the major segment is 616 − 56 = 560 cm². The major segment is also the major sector plus the triangle OAB: 462 + 98 = 560.
Notes
Major sector = πr² − minor sector = (360 − θ)/360 × πr². Major segment = πr² − minor segment.
Check yourself
A sector of a circle of radius 7 cm has angle 90°. Find the area of the corresponding major sector. (Take π = 22/7)
Answer: 115.5 cm²
Circle = 154. Minor sector = 154 / 4 = 38.5. Major sector = 154 − 38.5 = 115.5 cm². (Also 270/360 × 154 = 115.5.)
A circle has area 154 cm², and its minor segment has area 14 cm². Find the area of the major segment.
Answer: 140 cm²
Major segment = 154 − 14 = 140 cm².
A sector of a circle of radius 10 cm has angle 72°. Find the area of the corresponding major sector. (Take π = 3.14)
Answer: 251.2 cm²
Circle = 314. Minor sector = 314 / 5 = 62.8. Major sector = 314 − 62.8 = 251.2 cm².
Which formula gives the area of the major segment AQB?
- πr² − area of the minor segment APB — correct. Yes! The two segments together make the whole circle.
- (360 − θ)/360 × πr² − area of Δ OAB. The triangle OAB lies inside the major segment, so it must be added to the major sector, not taken away.
- θ/360 × πr². That is the area of the minor sector.