Question

find the area of the blue shaded region.
8 m
area = ? m²
61°
8 m

Explanation:

Answer:

To find the area of the blue shaded region, we need to calculate the area of the sector and subtract the area of the triangle.

Step 1: Calculate the area of the sector

The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is:

$$ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 $$

Given \( r = 8 \, \text{m} \) and \( \theta = 61^\circ \):

$$ \text{Area of sector} = \frac{61^\circ}{360^\circ} \times \pi \times (8)^2 $$

$$ \text{Area of sector} = \frac{61}{360} \times 64\pi $$

$$ \text{Area of sector} = \frac{3844}{360}\pi \approx \frac{3844}{360} \times 3.1416 \approx 33.7 \, \text{m}^2 $$

Step 2: Calculate the area of the triangle

The formula for the area of a triangle with two sides \( a \) and \( b \) and included angle \( \theta \) is:

$$ \text{Area of triangle} = \frac{1}{2}ab \sin \theta $$

Given \( a = 8 \, \text{m} \), \( b = 8 \, \text{m} \), and \( \theta = 61^\circ \):

$$ \text{Area of triangle} = \frac{1}{2} \times 8 \times 8 \times \sin 61^\circ $$

$$ \text{Area of triangle} = 32 \times \sin 61^\circ $$

$$ \sin 61^\circ \approx 0.8746 $$

$$ \text{Area of triangle} \approx 32 \times 0.8746 \approx 27.99 \, \text{m}^2 $$

Step 3: Calculate the area of the blue shaded region

Subtract the area of the triangle from the area of the sector:

$$ \text{Area of shaded region} = \text{Area of sector} - \text{Area of triangle} $$

$$ \text{Area of shaded region} \approx 33.7 - 27.99 \approx 5.71 \, \text{m}^2 $$

So, the area of the blue shaded region is approximately \( \boxed{5.71} \, \text{m}^2 \).