A 10-sided regular polygon is inscribed in a circle, as shown. The radius of the circle is approximately 8.1 centimeters. The length of each side of the polygon is 5 centimeters, and the length of the apothem is approximately 6.4 centimeters.
A figure.Short description, A figure.,Long description,
The figure shows a 10-sided polygon inscribed within a circle. The shaded region is one of the 10 sections that is between the outside of the 10-sided polygon and the edge of the circle.
Question
What is the approximate area of the shaded region, in square centimeters? Round the answer to the nearest tenth. Enter the answer in the box.
To find the area of the shaded region, we need to subtract the area of the regular polygon from the area of the circle.
The formula for the area of a regular polygon is A = (1/2)ap, where A is the area, a is the apothem, and p is the perimeter. In this case, the apothem is given as 6.4 cm, and the length of each side is given as 5 cm. Since there are 10 sides in the polygon, the perimeter is 10 times the length of each side, which is 10 * 5 = 50 cm.
So, the area of the regular polygon is A = (1/2)(6.4)(50) = 160 cm² (rounded to the nearest tenth).
The formula for the area of a circle is A = πr², where A is the area and r is the radius. In this case, the radius is given as approximately 8.1 cm.
So, the area of the circle is A = π(8.1)² ≈ 207.26 cm².
To find the area of the shaded region, we need to subtract the area of the regular polygon from the area of the circle:
Shaded area = Area of circle - Area of regular polygon = 207.26 - 160 ≈ 47.3 cm².
Therefore, the approximate area of the shaded region is 47.3 square centimeters.