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

Part 1)
`A=36(\pi-2)\ cm^2`

Part 2)
`P=6(\pi+2√(2))\ cm`

Explanation:

The picture of the question in the attached figure

Part 1) Find the area

we know that

The area of the shape is equal to the area of a quarter of circle minus the area of an isosceles right triangle

so

`A=(1)/(4)\pi r^(2)-(1)/(2)(b)(h)`

we have that the base and the height of triangle is equal to the radius of the circle

`r=12\ cm\\b=12\ cm\\h=12\ cm`

substitute

`A=(1)/(4)\pi (12)^(2)-(1)/(2)(12)(12)\\A=(36\pi-72)\ cm^2`

simplify

Factor 36

`A=36(\pi-2)\ cm^2`

Part 2) Find the perimeter

The perimeter of the figure is equal to the circumference of a quarter of circle plus the hypotenuse of the right triangle

The circumference of a quarter of circle is equal to

`C=(1)/(4)(2\pi r)=(1)/(2)\pi r`

substitute the given values

`C=(1)/(2)\pi (12)\\C=6\pi\ cm`

Applying the Pythagorean Theorem

The hypotenuse of right triangle is equal to

`AC=√(12^2+12^2)\\AC=√(288)\ cm`

simplify

`AC=12√(2)\ cm`

Find the perimeter

`P=(6\pi+12√(2))\ cm`

simplify

Factor 6

`P=6(\pi+2√(2))\ cm`