Question

find the area of the shaded segment of the circle.
the area of the shaded segment is \\(\square\\) \\(\text{m}^2\\).
(round to the nearest tenth as needed.)

Explanation:

Step1: Find the central angle of the segment

The total angle in a circle is \(360^\circ\). The unshaded major arc is \(300^\circ\), so the central angle of the segment (\(\theta\)) is \(360^\circ - 300^\circ = 60^\circ\). Convert this to radians or use the formula for the area of a sector and triangle. We'll use degrees here.

Step2: Calculate the area of the sector with central angle \(60^\circ\)

The formula for the area of a sector is \(A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2\), where \(r = 4\) m and \(\theta = 60^\circ\).

$$ A_{sector} = \frac{60^\circ}{360^\circ} \times \pi \times (4)^2 = \frac{1}{6} \times \pi \times 16 = \frac{8\pi}{3} \approx 8.3776 \text{ m}^2 $$

Step3: Calculate the area of the equilateral triangle (since two sides are radii and angle is \(60^\circ\), it's equilateral)

The formula for the area of an equilateral triangle with side length \(a\) is \(A_{triangle} = \frac{\sqrt{3}}{4} a^2\). Here, \(a = 4\) m.

$$ A_{triangle} = \frac{\sqrt{3}}{4} \times (4)^2 = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \approx 6.9282 \text{ m}^2 $$

Step4: Calculate the area of the segment

The area of the segment is the area of the sector minus the area of the triangle.

$$ A_{segment} = A_{sector} - A_{triangle} \approx 8.3776 - 6.9282 \approx 1.4494 \approx 1.4 \text{ m}^2 $$

(rounded to the nearest tenth)

Answer:

\(1.4\)