Question

The figures below are made out of circles, semicircles, quarter circles, and a square. Find the area and the perimeter of each figure and give your answers as a completely simplified exact value in terms of π (no approximations).

Answer 1

I.  ABCD is a square so  ∡BCD=90°
   |BC|²+|CD|² = |BD|²
   8² + 8² = |BD|²
     |BD|² = 8²·2
     |BD|  = √[8²·2]
     |BD|  = 8√2

II. Circular segment BCD is a quater of circle (because ∡BCD=90°)
R=8

So the lenght of arc BD:      $L_{BD}=\frac14\cdot 2\pi\cdot R = \frac12\cdot\pi\cdot8 = 4\pi$

Perimeter of the figure:    
                                       $P=|BC|+L_{BC} = 8\sqrt2+4\pi = 4(2\sqrt2+\pi)\ \text{in}$

The area of circular segment BCD: 
                                              $A_{BCD}=\frac14\cdot \pi R^2 =\frac14\cdot \pi\cdot 8^2 =\frac14\cdot \pi\cdot 64 =16\pi$
III. The area of triangle BCD:
                                                 $A_{\Delta BCD}=\frac12\cdot8\cdot8=32$
IV. The area of figure:
                                      $A=A_{BCD}-A_{\Delta BCD}=16\pi-32=16(\pi-2)\ \text{in}$

Answer 2

Answer:

A=16(pi-2),p=4(2 sqrt2+pi)

Step-by-step explanation: