A segment of a circle whose radius is 28cm subtends an angle of 150°. Calculate the area of the segment in two decimal places.
π * (28 cm)^2 * 150º / 360º = ? cm^2
R_scott has found the area of the SECTOR of the described circle.
A segment is the shape formed by a chord and its arc.
So from R_scott's answer of 980/3 π , we have to subtract the area of the
triangle.
We can use "area of triangle" = (1/2)(ab)sinθ
where a and be are two sides of a triangle and θ is the angle between them
so area of triangle = (1/2)(28)(28)sin 150
= 196
area of segment = 980/3 π - 196 cm^2 = appr 830.25 cm^2
To calculate the area of a segment of a circle, we need to find the area of the corresponding sector and subtract the area of the triangle formed by the radii of the sector.
First, let's find the area of the sector. The formula for the area of a sector is given by:
Area of sector = (θ/360) * π * r²
where θ is the central angle in degrees and r is the radius of the circle.
In this case, θ is 150° and r is 28cm. Plugging in these values:
Area of sector = (150/360) * π * (28)^2
Simplifying the expression:
Area of sector = (5/12) * 3.14159 * 784
Next, let's find the area of the triangle formed by the radii. The formula for the area of a triangle is given by:
Area of triangle = (1/2) * base * height
In this case, the base is the length of one of the radii, which is 28cm. The height can be calculated using the sine of the central angle as:
height = r * sin(θ)
Substituting the values, we get:
height = 28 * sin(150°)
Calculating the sine of 150°:
height = 28 * sin(2π/3) (converting 150° to radians)
height = 28 * √3/2
Simplifying the expression:
height = 14√3 cm
Plugging in the values in the formula for the area of the triangle:
Area of triangle = (1/2) * 28 * 14√3
Finally, to find the area of the segment, we subtract the area of the triangle from the area of the sector:
Area of segment = Area of sector - Area of triangle
Calculating the values:
Area of segment = [(5/12) * 3.14159 * 784] - [(1/2) * 28 * 14√3]
Now, simply evaluate the expression to get the area of the segment in two decimal places.