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 simpl…

Question

asked Jul 20, 2021 78.6k views

1 Answer

MJZ · Answer 1 · 2021-07-24T01:36:06+0000

Answer:

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

$P=6(\pi+2√(2))\ cm$