Answer:
- area: 27.8 square units
- perimeter: 21.2 units
Step-by-step explanation:
The area of a compound figure is the sum of the areas of its parts. The perimeter is the sum of the lengths of all of the edges.
Area
This compound figure is conveniently divided into a semicircle of radius 2.5 and a trapezoid with bases 5 and 7, and height 3.
Semicircle
The area of the semicircle is half the area of a circle with the same radius. It will be ...
A = 1/2πr²
A = 1/2π(2.5²) = 3.125π . . . . square units
Trapezoid
The area of the trapezoid is given by the formula ...
A = 1/2(b1 +b2)h
A = (1/2)(5 +7)(3) = 18 . . . . square units
Then the total area of the figure is ...
3.125π +18 ≈ 27.8 . . . . square units
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Perimeter
The perimeter will be the sum of the lengths of the straight line segments and the length of the semicircular arc.
Arc
The arc length is half the circumference of the circle, so is ...
arc = 1/2(2πr) = πr = 2.5π . . . . units
Diagonal segments
The figure is bounded by two congruent line segments that are each the hypotenuse of a triangle 1 unit wide and 3 units high. The Pythagorean theorem tells us that length is ...
diagonal length = √(1² +3²) = √10 . . . . units
The two diagonal sides have a total length of 2√10 units.
Horizontal segments
The figure is bounded by two congruent horizontal segments of length 1 unit each, and one horizontal segment of length 5 units. Their total length is ...
horizontal length = 1 + 1 + 5 = 7 . . . . units
The total perimeter is ...
perimeter = horizontal length + diagonal length + arc length
7 +2√10 +2.5π ≈ 21.2 . . . . units
