How to find the radius of a cone whose sector angle is 160 degree and the slant height is 2cm

Answer 1

To find the radius of a cone given the sector angle and slant height, follow these steps:

1. First, identify the given information:
- Sector angle (θ) = 160 degrees
- Slant height (l) = 2 cm

2. The sector angle of a cone is related to the cone's total surface area. The formula for the total surface area (A_total) of a cone is given by:
A_total = πr(l + r)
where r is the radius of the cone and l is the slant height of the cone.

3. Since the sector angle is 160 degrees, you know that the surface area of the whole cone is 360 degrees. Using this information, you can calculate the ratio of the given sector angle to the total surface area:
Ratio of sector angle to total area = θ / 360 = 160 / 360 = 4 / 9.

4. Next, use this ratio to find the total surface area of the cone. Since the given slant height is 2 cm, you can substitute the value of l and the ratio into the surface area formula to solve for the radius, r:
A_total = πr(l + r)
4/9 = [2πr(2 + r)] / [πr(2 + r)]
4/9 = 4 / (2 + r)
2 + r = 9
r = 9 - 2
r = 7 cm

5. Therefore, the radius of the cone is 7 cm.