☰ Contents · Geometry

Arc length and radian measure

Lessons 43 · 1 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
43

Length of an arc. The radian measure of an angle

Textbook: pp. 118–119
GoalFind the length of an arc of an n° central angle by l = πRn/180; meet the radian measure α = l/R and convert between degrees and radians.
New words
arc length l · yoy uzunligi lcentral angle · markaziy burchakradian · radianπ radians = 180° · π radian = 180°
Explanation

A whole circle of radius R has length 2πR and corresponds to 360°. So a 1° arc has length 2πR/360 = πR/180, and an n° arc has l = πRn/180. The ratio l/R depends only on the central angle, so it is taken as the radian measure of the angle: α = l/R. Here 180° = π radians, so an n° angle has radian measure α = πn/180. For example 90° = π/2, 60° = π/3 and 30° = π/6. In radian measure the arc length is very simple: l = αR.

Worked examples
R = 9, n = 40°: l = π·9·40/180 = 2π.
An angle of 3π/4 radians equals (3π/4)·(180°/π) = 135°. If R = 4 then l = (3π/4)·4 = 3π.
Class activity

“Clock hand”: take the tip of a minute hand at distance 10 cm from the centre. Find the length of the arc it draws in 20 minutes (the angle is 120°).

Practice
1
What is the length of a 150° arc of a circle of radius 12 cm?
2
How many radians is 120°?
3
The central angle is 2 radians and the radius is 7 m. What is the arc length?
4
Why does the radian measure not depend on the radius?