Question

Find a formula for the area A of a circular sector in terms of its radius r and arc length s

Answer 1

\[s=r \theta\] and
\[A=1/2r ^{2}\theta\] then\[A=1/2rs\]

Answer 2

The ratio of the sector area to the circle area is the same as the ratio of the arc length to the circle circumference.\[\frac{A_{sector}}{A_{circle}} = \frac{s}{C_{circle}}\]\[A_{circle} = \pi r^{2}\]\[C_{circle} = 2\pi r\]Cross-multiplying we get \[A_{sector} = \frac{s A_{circle}}{C_{circle}} = \frac{s \pi r^{2}}{2\pi r} = \frac{sr}{2}\]