☰ Contents · Geometry

Area of a disc and its parts

Lessons 44–45 · 2 lessons · B. Xaydarov, E. Sariqov, A. Qo‘chqorov. Geometry, Grade 9 (textbook for general secondary schools), revised 4th edition. Huquq va Jamiyat Publishing, Tashkent, 2019
45

Areas of parts of a circle

Textbook: pp. 122–123
GoalKnow the definitions of a sector and a segment and their area formulas (S = πR²n/360, S = ½Rl, segment = sector ∓ triangle) and apply them.
New words
sector · sektorsegment · segmentS = πR²n/360 · S = πR²n/360arc length l · yoy uzunligi l
Explanation

A sector is the part of a disc bounded by an arc and the two radii to its endpoints. A sector with an n° arc is n/360 of the disc, so its area is S = πR²·n/360, which also equals ½·R·l (l is the arc length). A segment is the part bounded by an arc and its chord. For a segment smaller than a half-disc, the area is the sector minus the triangle: S = πR²n/360 − S(AOB). For a segment larger than a half-disc, the triangle's area is added. For a 90° arc the triangle has area R²/2.

Worked examples
R = 6, n = 60°: S = π·36·60/360 = 6π.
R = 4, n = 90°: segment = 4π − ½·4·4 = 4π − 8.
Class activity

“Pizza slice”: divide a paper disc into 8 equal sectors; check that one sector is 1/8 of the disc and has an angle of 45°.

Practice
1
What is the area of a 120° sector of a disc of radius 9?
2
A sector has arc length 6π and radius 5. Find its area (S = ½Rl).
3
What is the area of the segment cut off by a 90° arc in a disc of radius 2?
4
Why is the area of a 90° sector a quarter of the disc?