Find the area of the entire shape.

Question

ABCD is a straight line. ABE is a sector of a circle with center B. CED is a sector of a circle with center C. Angle ABE is a right angle. The length of AB is r and angle ACE is π/4 radians. If r=10cm, find the area of the shape.

Length of $CD= √2 r$, Angle $ECD= 3π/4,$ Area of shape= Area of AEB + Area of EBC + Area of ECD = 25π + 50 + 75π = 364cm^2 (to 3s.f.)

Answer 1

I think, your solution is right.

I got the same result: $$\frac{\pi r^2}{4}+\frac{r^2}{2}+\frac{3\pi(r\sqrt2)^2}{8}=\left(\pi+\frac{1}{2}\right)r^2=100\pi+50.$$

Answer 2

Hints: $\left| \bar{AB} \right | = \left| \bar{BE} \right |, \\ \sin\frac{\pi}{4} = \frac{\left| \bar{BE} \right |}{\left| \bar{CE} \right |}, \\ \left| \bar{CE} \right | = \left| \bar{CD} \right |, \\ \left| \bar{BC} \right |^2 = \left| \bar{CE} \right |^2 - \left| \bar{BE} \right |^2.$