Steve can you please show me how to do this problem?
What is the area of the shaded region in the given circle in terms of pi and in the simplest form?
the picture shown is of a circle with a triangle from 1 oclock to 3 oclock with a 60 deg opening. the side length of the triangle is 18m and the little part where the triangle and the edge of the circle meet between 1 and 3 oclock is the only part not shaded. The choices we have as answers are...
A. (270pi+54 sqrt3)m^2
B. (216pi+54 sqrt3)m^2
C. (270pi+108 sqrt3)m^2
D. (270pi+81 sqrt3)m^2
I hope you understand this explanation.
did u smash
To find the area of the shaded region, we first need to calculate the area of the entire circle and then subtract the area of the triangle.
The area of a circle is given by the formula: A = πr^2, where A is the area and r is the radius.
Given that the triangle has a 60-degree opening, we can find the radius of the circle using the following steps:
1. The length of the side of the triangle is 18m.
2. The triangle is an equilateral triangle because all angles are 60 degrees.
3. For an equilateral triangle, the radius of the circle circumscribing the triangle is given by the formula: r = s√3/3, where s is the side length of the triangle.
4. Substituting the given value, we get r = (18√3)/3 = 6√3 m.
Now that we have the radius of the circle, we can find its area:
A_circle = πr^2
= π(6√3)^2
= 36π(3)
= 108π m^2.
Next, we need to find the area of the triangle. The area of an equilateral triangle is given by the formula: A_triangle = (s^2√3)/4, where s is the side length of the triangle.
A_triangle = (18^2√3)/4
= (324√3)/4
= 81√3 m^2.
Finally, we subtract the area of the triangle from the area of the circle to find the area of the shaded region:
Area of shaded region = A_circle - A_triangle
= 108π - 81√3
= 27(4π - 3√3)
= 27(4π - 3√3)m^2.
Comparing the result to the answer choices, we can see that the correct answer is C. (270π + 108√3)m^2.
Thank you Steve...
since the triangle is equilateral, the radius of the circle is 18.
so, your area is 5/6 of a circle plus the area of the triangle, which is 18^2/4 √3 = 81√3