Find the exact area of the shaded region.
An image of a circle with a triangle inside of the circle. The triangle has a 120 degree angle with a side length of 24m.
(1 point)
Responses
(192symbol pi – 144)m²
(192 Image with alt text: symbol pi – 144)m²
(192symbol pi – 144Start Root 3 End Root)m²
(192 Image with alt text: symbol pi – 144 Image with alt text: Start Root 3 End Root )m²
(8symbol pi – 144Start Root 3 End Root)m²
(8 Image with alt text: symbol pi – 144 Image with alt text: Start Root 3 End Root )m²
none of the above
To find the area of the shaded region, we need to subtract the area of the triangle from the area of the circle.
The area of the triangle can be found using the formula: A = (1/2)bh, where b is the base and h is the height. Since we have a 120 degree angle and a side length of 24m, the base of the triangle is 24m and the height can be found using the formula: h = b * sin(angle), where sin(angle) is the sine of the angle. In this case, sin(120) = √3/2. Therefore, the height is (24)(√3/2) = 12√3.
The area of the triangle is (1/2)(24)(12√3) = 144√3.
The area of the circle can be found using the formula: A = πr^2, where r is the radius. Since we know the side length of the square is 24m, the radius can be found using the formula: r = (24√3)/3, which simplifies to 8√3.
The area of the circle is π(8√3)^2 = 64π(3) = 192π.
Therefore, the area of the shaded region is 192π - 144√3, which is approximately 655.83 square meters.