The vertical angle of a cone is 60°and its slant height is 30cm. Find the radius of the base
To find the radius of the base of the cone, we can use the formula for the slant height of a cone in terms of the radius and height:
slant height = √(radius^2 + height^2)
In this problem, the slant height is given as 30 cm. We need to find the radius.
First, we can find the height of the cone using trigonometry. Since the vertical angle is given as 60°, and the slant height is the hypotenuse of a right triangle, we can use sine:
sin(vertical angle) = height / slant height
Substituting the given values, we get:
sin(60°) = height / 30
√3 / 2 = height / 30
Simplifying the equation, we get:
height = (30 * √3) / 2
height = 15√3 cm
Now that we have the height and slant height, we can find the radius of the base using the Pythagorean theorem:
(radius)^2 + (height)^2 = (slant height)^2
Substituting the given values, we get:
(radius)^2 + (15√3)^2 = 30^2
(radius)^2 + 225 = 900
(radius)^2 = 900 - 225
(radius)^2 = 675
Taking the square root of both sides, we get:
radius = √675
Hence, the radius of the base of the cone is √675 cm.