Question

find the area of the shaded region.
10 cm
70°
10 cm
$a_{sector} = \frac{sector\\ angle}{360°}\pi r^2$
$a_{triangle} = \frac{1}{2} r^2 \sin(angle)$
$a_{shaded} = ? cm^2$
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the area of the sector

The formula for the area of a sector is \( A_{\text{sector}} = \frac{\text{sector angle}}{360^\circ} \pi r^2 \). Here, the sector angle is \( 70^\circ \) and the radius \( r = 10 \) cm. Plugging in the values:

$$ A_{\text{sector}} = \frac{70^\circ}{360^\circ} \times \pi \times (10)^2 = \frac{70}{360} \times 100\pi = \frac{7000\pi}{360} \approx \frac{7000 \times 3.1416}{360} \approx 61.0865 \text{ cm}^2 $$

Step2: Calculate the area of the triangle

The formula for the area of the triangle is \( A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\text{angle}) \). With \( r = 10 \) cm and angle \( 70^\circ \):

$$ A_{\text{triangle}} = \frac{1}{2} \times (10)^2 \times \sin(70^\circ) = 50 \times \sin(70^\circ) \approx 50 \times 0.9397 \approx 46.985 \text{ cm}^2 $$

Step3: Find the area of the shaded region

The area of the shaded region is the area of the sector minus the area of the triangle:

$$ A_{\text{shaded}} = A_{\text{sector}} - A_{\text{triangle}} \approx 61.0865 - 46.985 \approx 14.1015 \approx 14.10 \text{ cm}^2 $$

Answer:

\( 14.10 \)