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Math  /  Geometry

QuestionA rectangle is placed around a semicircle as shown below. The width of the rectangle is 7 cm . Find the area of the shaded region. Use the value 3.14 for π\pi, and do not round your answer. Be sure to include the correct unit in your answer. □\square cm cm2\mathrm{cm}^{2} cm3\mathbf{c m}^{3}

Studdy Solution

STEP 1

1. The semicircle is perfectly inscribed in the rectangle, meaning its diameter equals the width of the rectangle.
2. The width of the rectangle is 7 7 cm.
3. The semicircle is the shaded region.

STEP 2

1. Determine the radius of the semicircle.
2. Calculate the area of the semicircle.
3. State the area of the shaded region.

STEP 3

Determine the radius of the semicircle:
The diameter of the semicircle is equal to the width of the rectangle, which is 7 7 cm. Therefore, the radius r r of the semicircle is:
r=Diameter2=7 cm2=3.5 cm r = \frac{\text{Diameter}}{2} = \frac{7 \text{ cm}}{2} = 3.5 \text{ cm}

STEP 4

Calculate the area of the semicircle:
The formula for the area of a semicircle is:
Area of semicircle=12πr2 \text{Area of semicircle} = \frac{1}{2} \pi r^2
Substitute the known values:
Area of semicircle=12×3.14×(3.5 cm)2 \text{Area of semicircle} = \frac{1}{2} \times 3.14 \times (3.5 \text{ cm})^2
=12×3.14×12.25 cm2 = \frac{1}{2} \times 3.14 \times 12.25 \text{ cm}^2
=19.215 cm2 = 19.215 \text{ cm}^2

STEP 5

State the area of the shaded region:
The shaded region is the area of the semicircle, which is:
19.215 cm2 \boxed{19.215 \text{ cm}^2}

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