Answer:
- A = (18π +72) cm²
- P = (12π +12) cm
Step-by-step explanation:
area
The area of a semicircle is ...
A = 1/2(πr²)
For a radius of 6 cm, the (top) semicircle area is ...
A = (1/2)π(6 cm)² = 18π cm²
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We notice that the area of the bottom-most semicircle exactly fills the empty space next to it. That is, the area of the bottom part of the figure is exactly the area of 2 squares of side length 6 cm.
rectangle area = 2(6 cm)² = 72 cm²
Then the total area of the figure is ...
(72 +18π) cm²
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perimeter
The perimeter of a semicircle of radius r is ...
P = 1/2(2πr) = πr
So, the length of arc AE is ...
AE = π(6 cm) = 6π cm
The two semicircles at the bottom of the figure have a total arc length exactly equal to the circumference of a circle 6 cm in diameter. That is ...
C = πd
C = π(6 cm) = 6π cm
The two straight sides of the figure have a total length equal to twice the side length of each square:
P = 2s = 2(6 cm) = 12 cm
Then the perimeter of the figure is ...
AE +(AB+DE) +BCD
= 6π cm + 12 cm + 6π cm
= (12 +12π) cm
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Additional comment
Usually, "completely simplified" means that parentheses are eliminated from the expression and like terms are combined. While the perimeter expression shown in the problem statement is numerically correct, it does not qualify as being "completely simplified". (You may prefer to write the π-term first. Often the answer checker will choke on parentheses, and it may prefer the value have no spaces. YMMV)
