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2 years ago

Find the area and perimeter of the compound figure below. (6 points)

Mathematics

1 answer:

fenix001 [56]2 years ago

7 0

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.

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

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

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