A circular sector has two straight sides of 15 m each and a curved outer edge of 37.6 m. Find its area.

Step-by-step solution

1. 1

   Identify the shape from the given lengths. Two equal straight sides meeting a single curved edge describe a circular sector - a pizza slice. The equal sides are radii, so $r=15\text{ m}$, and the curved edge is the arc, $L=37.6\text{ m}$.

2. 2

   Choose the formula that uses arc length directly. The central angle is not given, but a sector obeys $A=\frac{rL}{2}$. This follows from $A=\frac12 r^2\theta$ and $L=r\theta$: substituting $\theta=L/r$ turns $\frac12 r^2\theta$ into $\frac12 rL$, so no angle is ever needed.

3. 3

   Substitute the two given lengths. $A=\frac{15\cdot 37.6}{2}=\frac{564}{2}$.

4. 4

   Compute the area. $A=282\text{ m}^2$.

5. 5

   Cross-check through the central angle. $\theta=\frac{L}{r}=\frac{37.6}{15}\approx 2.5067$ radians, about $143.6^\circ$. Then $\frac12 r^2\theta=\frac12(225)(2.5067)=282.0\text{ m}^2$, the same value, and the angle is plausibly under $180^\circ$ for a slice.

6. 6

   Note the perimeter, which is often asked alongside. The boundary is two radii plus the arc: $15+15+37.6=67.6\text{ m}$.