Use radians before applying arc and sector formulas. Subtract the triangle from the corresponding sector.
Notes and key methods
One radian subtends an arc equal in length to the radius. A full turn is 2π radians, so degrees convert by multiplying by \(\displaystyle \frac{\pi }{180}\).
With θ in radians, arc length is \(\displaystyle r\theta\) and sector area is \(\displaystyle \frac{r^{2}\theta }{2}.A\) sector perimeter also includes its two straight radii. Check whether a stated angle describes the minor or major sector.
For 0<θ<π, a minor segment has area \(\displaystyle \frac{r^{2}(\theta - \sin \theta )}{2}\): subtract the isosceles triangle from the sector with the same arc.
The chord is a straight line and the arc is curved. Use \(\displaystyle 2r\sin (\frac{\theta }{2})\) for the chord, and \(\displaystyle r\theta\) for the arc. In a composite region, label every area before adding or subtracting it.
Check the conditions
Angles in \(\displaystyle r\theta\) and \(\displaystyle \frac{r^2\theta}{2}\) must be radians.
The radius and all lengths use consistent units.
Use the minor segment and its stated angle.
Use radians in the sector area formula.
Three worked examples
Example 1 · Radians, arcs and sectors
A sector has radius \(\displaystyle r=7 \mathrm{cm}\) and angle \(\displaystyle \frac{\pi}{2}\) radians.
Find the exact arc length and sector area.
Find the exact sector perimeter.
Find the combined length of the two straight radii.
Find the fraction of a full turn occupied by the sector.
Show the reasoning
(a)
\(\displaystyle s = \frac{7 \pi}{2}\) and \(\displaystyle A = \frac{49 \pi}{4}\) use the radian angle.
Example 2 · Segments and composite circular regions
A circle has radius 7 cm. Two radii enclose an angle of \(\displaystyle \frac{\pi }{2}\) radians; consider the minor segment between their chord and arc.
Find the exact minor-segment area.
Find the exact chord and arc lengths.
Find the squared chord length.
Find the arc length divided by π.
Show the reasoning
(a)
Subtract triangle area \(\displaystyle \frac{49}{2}\) from sector area \(\displaystyle \frac{49 \pi}{4}\).