Find the volume of the surface of the figure generated by rotating a 30-60-90 triangle about its longest leg if the hypotenuse is 8in

Question

in = inches. This makes a cone right? How do I find the other sides and what is the formula for surface area?

Answer 1

By longest leg I'm assuming you mean the hypotenuse.

Rotating the triangle makes a cone.

First, find the radius and altitude.

The side opposite the 30° angle is half the hypotenuse, or 4 in. This is the radius of the cone.

Use the Pythagorean Theorem to find the other leg which will be the altitude.

8^2=4^2+h^2

64=16+h^2

h^2=48

h=4√3

V=(1/3)∏r^2h

V=(1/3)(∏)(4^2)(4√3)

V=(1/3)(∏)(16)(4√3)

V=(1/3)(3.14159)(16)(6.9282)

V=(1.04719)(110.8512)

V=116.08227 cubic inches (round as needed)