Answer :
Final answer:
To calculate the side length of a regular pentagon with a 25-inch radius circle, we use trigonometry. Specifically, we use the cosine of a 36° angle in a resulting right triangle, which yields a side length of approximately 40.45 inches.
Explanation:
The student has asked for the length of the sides of a regular pentagon inscribed in a circle with a radius of 25 inches. To solve this, we can use the concept of trigonometry and the properties of a regular pentagon. The central angle of a pentagon is 360°, divided by the number of sides, which is 5, resulting in a 72° angle. Each segment from the center to a vertex of the pentagon forms an isosceles triangle with the radius as its sides and the pentagon side as its base.
Using trigonometry, the length of a side 's' of the pentagon can be determined from such a triangle:
- Split the isosceles triangle into two right triangles by drawing a perpendicular from the center to one side. Each right triangle will have a base 's/2' (half of the pentagon side) and a hypotenuse equal to the radius 'r' (25 inches).
- The angle θ at the center for each of these right triangles is half of the 72°, which is 36°.
- Use the cosine function, where cos(θ) = adjacent/hypotenuse, which translates to cos(36°) = (s/2)/25.
- Rearrange to find the side length: s = 2 × 25 × cos(36°).
- Calculate the value: s ≈ 2 × 25 × 0.8090 ≈ 40.45 inches.
The length of each side of the pentagon is therefore approximately 40.45 inches.
