Pure Mathematics 1
4. Circular Measures
Written by: Tharun Athreya
Formatted by: Dhyaneshwaran V
Index
4.1 Radians
- An arc equal in length to the radius of a circle subtends an angle of 1 radian at the centre.
- 2𝜋 radians = 360°
𝜋 radians = 180° - To change from degrees to radians, multiply by $$\frac{\pi}{180}$$
- To change from radians to degrees, multiply by $$\frac{180}{\pi}$$
4.2 Length of an arc
- If an arc subtends an angle of 𝜃 radians at the centre, the length of the arc is r𝜃.
4.3 Area of a sector
- To find the formula for the area of a sector, we use the ratio: $$ \frac{\text{area of sector}}{\text{area of circle}} = \frac{\text{angle in the sector}}{\text{complete angle at the centre}}$$
- To find the formula for the area of a sector, we use the ratio: $$\frac{\text{area of sector}}{\pi r^2} = \frac{\theta}{2\pi}$$
- Area of sector = $$\frac{\theta}{2\pi} \times \pi \times r^2
$$
$$\frac{1}{2} \times \pi \times r^2 \times \theta
$$
Exam tip: Check the mode (degrees/radians) of your calculator before using it.