Question

An equilateral triangle has a side length of $12 \sqrt{3}.$ The triangle is rotated about one of its altitudes as an axis, sweeping out a cone in space. Find the volume of the cone.

Answer 1

The volume of the cone is 254.85 cubic units

How to determine the volume of the cone?

The side length of the equilateral triangle is given as:

l = 12√3

Calculate the radius (r) of the cone using:

r = l/2

r = 12/2√3

r = 6√3

Calculate the height of the cone using:

$h = \sqrt{l^2 + (l/2)^2$

This gives

$h = \sqrt{(12\sqrt 3)^2 + (6\sqrt3)^2$

Evaluate

$h = \sqrt{540$

h = 22.24

The volume of the cone is then calculated as:

V = 1/3πr²h

This gives

V = 1/3π * (6√3)² * 22.24

Evaluate

V = 254.85

Hence, the volume of the cone is 254.85 cubic units

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