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).
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: [tex]L_{BD}=\frac14\cdot 2\pi\cdot R = \frac12\cdot\pi\cdot8 = 4\pi [/tex]
Perimeter of the figure: [tex]P=|BC|+L_{BC} = 8\sqrt2+4\pi = 4(2\sqrt2+\pi)\ \text{in}[/tex]
The area of circular segment BCD: [tex]A_{BCD}=\frac14\cdot \pi R^2 =\frac14\cdot \pi\cdot 8^2 =\frac14\cdot \pi\cdot 64 =16\pi[/tex] III. The area of triangle BCD: [tex]A_{\Delta BCD}=\frac12\cdot8\cdot8=32[/tex] IV. The area of figure: [tex]A=A_{BCD}-A_{\Delta BCD}=16\pi-32=16(\pi-2)\ \text{in}[/tex]
