Answer :
To find the area between the two radii forming a 30-degree angle at the center of a circle with a radius of 10 cm, you can follow these steps:
1. **Find the Area of the Sector:**
- The area of the entire circle with a radius of 10 cm can be calculated using the formula:
Area of a circle = π * radius^2 = 3.14 * 10^2 = 314 square centimeters.
- Since the angle at the center is 30 degrees, the area of the sector formed by the two radii is a fraction of the area of the entire circle. The fraction is 30 degrees out of 360 degrees, so the area of the sector is (30/360) * 314 = 26.2 square centimeters.
2. **Subtract the Area of the Triangle:**
- The area between the two radii is the area of the sector minus the area of the triangle formed by the radii.
- The triangle formed by the two radii and the chord connecting them is an equilateral triangle with each side being 10 cm (the radius of the circle).
- To find the area of an equilateral triangle, you can use the formula:
Area of an equilateral triangle = (√3 / 4) * side^2 = (√3 / 4) * 10^2 = 25√3 square centimeters ≈ 43.3 square centimeters.
3. **Calculate the Final Area Between the Radii:**
- Subtract the area of the equilateral triangle from the area of the sector to find the area between the two radii:
26.2 - 43.3 ≈ -17.1 square centimeters.
Therefore, the correct answer is not among the options provided. The area between the two radii is approximately -17.1 square centimeters.
