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

The area of the figure be 137.76 units and perimeter of the figure be 42.39 units .

Explanation:

Formula

`Perimeter\ of\ a\ semi\ circle = (2\pi\ r)/(2)`

`Area\ of\ a\ circle = (\pi\ r^(2))/(2)`

Where r is the radius of circle .

As the figure given in the question consist of one big and Three small semicircle .

As

`Radius\ of\ big\ semicircle = (AB+BC+CD)/(2)`

As AB = BC = CD = 6

`Radius\ of\ big\ semicircle = (18)/(2)`

= 9 units

`Radius\ of\ small\ semicircle = (3)/(2)`

= 1.5 units

`\pi = 3.14`

`Perimeter\ of\ figure = Perimeter\ of\ Big\ semicircle + Perimeter\ of\ three\ semicircle`

`Perimeter\ of\ figure = 9* 3.14+3 * 3.14* 1.5`

= 28.26 + 14.13

= 42.39 units

`Area\ of\ figure = Area\ of\ Big\ semicircle + Area\ of\ three\ semicircle`

`= (3.14* 9* 9)/(2) + 3 (3.14* 1.5* 1.5)/(2)`

Area of the figure = 127.17 + 10.59

= 137.76 units

Therefore the area of the figure be 137.76 units and perimeter of the figure be 42.39 units .

Answer 2

Total area of shaded region is
`54 \pi}`

Total perimeter of figure is
`18 \pi`

Explanation:

Area of semi circle =
`(1)/(2) \pi r^2`

Perimeter of semi circle =
`\pi r`

Consider the given figure , it has 4 semi-circles with diameter AD , AB, BC, CD.

We have to determine the shaded area ,

First consider , Semi circle with diameter AD,

diameter AD = 6 + 6 + 6 = 18 cm

Radius is half of diameter.

Radius = 9 cm

Thus, area of semi circle with diameter 18 cm is,

Area of semi circle =
`(1)/(2) \pi (9)^2=(81 \pi)/(2)` .....(1)

Area of 3 semi circles with diameter 6 cm = 3 × Area of semi circles with diameter 6 cm

diameter = 6 cm ⇒ Radius = 3 cm

Area of semi circle =
`(1)/(2) \pi (3)^2=(9 \pi)/(2)` .....(2)

Area of 3 semi circles with diameter 6 cm = 3 × Area of semi circles with diameter 6 cm

`=(3 * 9 \pi)/(2)`

`=(27 \pi)/(2)` ..........(3)

Total area of shaded region = Area of semi circle with diameter 18 cm + Area of 3 semi circles with diameter 6 cm

=
`(81\pi)/(2)+(27 \pi)/(2)`

=
`((81+27) \pi)/(2)=(108 \pi)/(2)`

Thus, Total area of shaded region is
`54 \pi}`

Perimeter of the semi circle with radius 9 cm

Perimeter of semi circle =
`\pi r=9 \pi`

Perimeter of the one semi circle with radius 3 cm

Perimeter of semi circle =
`\pi r=3 \pi`

Perimeter of the 3 semi circle with radius 3 cm = 3 × Perimeter of the one semi circle with radius 3 cm

`=9 \pi`

Total perimeter of figure is
`9 \pi+9 \pi=18 \pi`