Answer:
To find the area of the shaded segment of the circle, we need to first find the area of the sector formed by the 60 degrees angle, and then subtract the area of the triangle formed by the radii and the chord.
The formula for the area of a sector of a circle is:
A_sector = (θ/360) x πr^2
where θ is the central angle in degrees, and r is the radius.
Substituting the given values, we get:
A_sector = (60/360) x π x 18^2
A_sector = 113.04 cm^2
Next, we need to find the area of the triangle formed by the radii and the chord. To do this, we need to first find the length of the chord using the formula:
c = 2r sin(θ/2)
where θ is the central angle in radians.
Converting the given 60 degrees angle to radians, we get:
θ = 60 x (π/180) = π/3 radians
Substituting the values, we get:
c = 2 x 18 x sin(π/6)
c = 18√3 cm
Now, we can find the area of the triangle using the formula:
A_triangle = (1/2) x base x height
where the base is the chord length c, and the height is half the length of the chord, which is:
h = r - r cos(θ/2)
Substituting the values, we get:
h = 18 - 18 cos(π/6)
h = 18 - 9√3 cm
Therefore, the area of the triangle is:
A_triangle = (1/2) x 18√3 x (18 - 9√3)
A_triangle = 243 cm^2
Finally, the area of the shaded segment is the difference between the area of the sector and the area of the triangle:
A_shaded = A_sector - A_triangle
A_shaded = 113.04 - 243
A_shaded = -129.96
However, the result is negative because the triangle is larger than the sector in this case. Therefore, there is no shaded area in this situation.
Explanation: