Question

A rectangle is placed around a semicircle as shown below. The length of the rectangle is 4mm
. Find the area of the shaded region.

Answer 1

Answer on Question #80656 – Math – Geometry

Question

A rectangle is placed around a semicircle as shown below. The length of the rectangle is 4mm. Find the area of the shaded region.

Given:

The length of the rectangle: $a = 4\,\mathrm{mm}$

Solution

The area of the shaded area is equal to the difference between the areas of the rectangle and the semicircle inscribed in it:

$$S = S_{\text{Rect}} - S_{\text{Semicircle}}$$

semicircle area:

$$S_{\text{Semicircle}} = \frac{\pi d^2}{8}$$

length:

$$a = d;$$

width:

$$b = \frac{d}{2};$$

Area of the rectangle:

$$S_{\text{Rect}} = a b;$$ $$S_{\text{Rect}} = \frac{d^2}{2}$$ $$S = \frac{d^2}{2} - \frac{\pi d^2}{8} = \frac{d^2}{8} (4 - \pi) = 1.72\,\mathrm{mm}^2$$

Answer:

area of the shaded region: $1.72\,\mathrm{mm}^2$

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