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).

Answer 1

Part 1)

A= 36(π-2)cm^2

P= 6(π+2√2)cm

Answer 2

Figure 1)

a) The area is A = 36( π - 2) ст²

b) The perimeter is P = 6 (π + 2√2)
`cm^2`

Figure 2)

a) The area is A = 576 cm²

b) The perimeter is P = 24(π+2)
`cm^2`

Figure 1)

a) To find the area of the figure 1 we have to do the substraction

The area of figure 1 is equivalent to the area of a triangle less the area of a quarter circle.

The area of quarter circle is equal to

A = 2

we have

r = 12 cm

Put the value r as given

A = (12)2

A = 36
`cm^2`

Area of a triangle formula is

A = (b)(h)

Given information,

b=12 cm

h = 12 cm

substitute

A = (12) (12)

A = 72 cm²

therefore

The area of the figure is

A = (36π - 72)
`cm^2`

Simplify

A = 36(π -2)
`cm^2`

b)

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

The perimeter of a quarter of circle is formula

C = 2πr

simplify

C = r

we have

r=12 cm

substitute

C = (12)

C = 67 cm

Find the length side AC

Applying the Pythagorean Theorem

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

`AC^2 = 144+ 144\\AC^2 = 288\\AC = √(288)`

AC = 12√2 cm

P = (6π +12√2) cm

Taking 6 as a common multiplier we get

P = 6(π+2√2)

The perimeter of the figure 1 is

P = 6(π+2√2) cm

Part 2)

a) As we can see,

The area of a semicircle plus the area of a square less the area of a semicircle equals the area of figure 2.

The figure's area and the square's area are equal.

A =
`24^2`

A = 576 cm²

b) Find the perimeter of the figure 2

we know that

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

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

P = 2(AB) + π D

P = 2(24) + π(24)

P = 48 + π(24)

Take out 24

P = 24 (2 + π ) cm