Question

For the figures below, assume they are made of semicircles, quarter circles and squares. For each shape, find the area and perimeter. Give your answer as a completely simplified exact value in terms of π (no approximations).

b

Answer 1

Part 1)

a) The area is
`A=36(\pi-2)\ cm^2`

b) The perimeter is
`P=6(\pi+2√(2))\ cm`

Part 2)

a) The area is
`A=576\ cm^2`

b) The perimeter is
`P=24(\pi+2)\ cm`

Explanation:

see the attached figure to better understand the problem

Part 1)

a) Find the area of the figure N 1

we know that

The area of the figure N 1 is equal to the area of a quarter circle minus the area of the triangle

Find the area of the quarter circle

The area of quarter circle is equal to

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

we have

`r=12\ cm`

substitute

`A=(1)/(4)\pi (12)^(2)`

`A=36\pi\ cm^2`

Find the area of the triangle

The area of triangle is equal to

`A=(1)/(2)(b)(h)`

we have

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

substitute

`A=(1)/(2)(12)(12)`

`A=72\ cm^2`

therefore

The area of the figure is

`A=(36\pi-72)\ cm^2`

Simplify

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

b) Find the perimeter of figure N 1

we know that

The perimeter of the figure N 1 is equal to the circumference of a quarter circle plus the side AC of triangle

Find the circumference of a quarter circle

The perimeter of a quarter of circle is equal to

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

simplify

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

we have

`r=12\ cm`

substitute

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

`C=6\pi\ cm`

Find the length side AC

Applying the Pythagorean Theorem

`AC^2=12^2+12^2`

`AC^2=288`

`AC=√(288)\ cm`

simplify

`AC=12√(2)\ cm`

The perimeter of the figure is

`P=(6\pi+12√(2))\ cm`

simplify

`P=6(\pi+2√(2))\ cm`

Part 2)

a) Find the area of the figure N 2

we know that

The area of the figure N 2 is equal to the area of a semicircle plus the area of a square minus the area of semicircle

so

The area of the figure is equal to the area of the square

`A=24^2\\A=576\ cm^2`

b) Find the perimeter of the figure N 2

we know that

The perimeter of the figure N 2 is equal to the length side AB plus the length side DC plus the circumference of two semicircles

so

The perimeter of the figure N 2 is equal to two times the length side AB plus the circumference of one circle

`P=2(AB)+\pi D`

`P=2(24)+\pi (24)`

`P=(48+24\pi)\ cm`

simplify

`P=24(\pi+2)\ cm`