Find the central angle of the polygon: To find the central angle, we need to divide the total circumference of the circle by the number of sides of the polygon. The formula for the circumference of a circle is 2πr, where r is the radius. The central angle of the polygon, in degrees, is given by 360°/n, where n is the number of sides.
In this case, the circumference of the circle is 2π * 16 = 32π, and the number of sides of the polygon is 7. The central angle is 360°/7 = 51.43°.
Find the length of the chord that corresponds to each central angle: The formula for the length of a chord of a circle is 2r * sin(θ/2), where r is the radius and θ is the central angle in radians. To convert degrees to radians, we use the formula radians = degrees * (π/180).
In this case, the central angle is 51.43°, so in radians, it is 51.43° * (π/180) = 0.897 radians. The length of the chord corresponding to this central angle is 2 * 16 * sin(0.897/2) = 15.26 cm.
Find the area of the segment: To find the area of the segment, we subtract the area of the triangle from the area of the sector that corresponds to the central angle. The formula for the area of a sector is (θ/360) * πr^2, and the formula for the area of a triangle is (1/2)bh, where b is the length of the chord and h is the height from the center of the circle to the chord.
The area of the sector is (51.43°/360) * π * 16^2 = 38.27 cm^2. The height of the triangle is the radius of the circle, which is 16 cm, and the length of the chord is 15.26 cm. The area of the triangle is (1/2) * 15.26 * 16 = 122.08 cm^2. The area of the segment is the difference between the area of the sector and the area of the triangle, or 38.27 - 122.08 = 83.81 cm^2.
So the area of the segment of the circle is 83.81 cm^2.