Question

The figures below are based on semicircles and squares. Find the perimeter and the area of each shape. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

Part 1) The area of the figure is
`A=18(\pi +4)\ cm^2`

Part 2) The perimeter of the figure is
`P=12(\pi +1)\ cm`

Explanation:

Part 1) Find the area

we know that

The area of the figure is equal to the area of semicircle with diameter AE, plus the square CDEF, plus the area of square ABCF, minus the area of semicircle of diameter BC plus the area of semicircle of diameter CD

This is equivalent to the area of semicircle with diameter AE plus two times the area of square CDEF

so

`A=(1)/(2)\pi (6)^(2) +2(6^2)`

`A=(18\pi +72)\ cm^2`

Simplify

Factor 18

`A=18(\pi +4)\ cm^2`

Part 2) Find the perimeter of the figure

we know that

The perimeter of the figure is equal to the circumference of semicircle with diameter AE plus two times the segment ED plus the circumference of the circle with diameter CD

so

`P=(1)/(2)\pi(12)+2(6)+\pi (6)`

`P=6\pi+12+6\pi\\P=(12\pi +12)\ cm`

Simplify

Factor 12

`P=12(\pi +1)\ cm`