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How to Find the Volume of a Cone Given a Half Angle

A cone is a three-dimensional object having a circular base and vertex. Any cone is characterized by the radius of the circular base and the height, which is the distance from the vertex to the center of the circle. The half angle refers to the angle between the height and the side line of the cone. These two lines and the cone radius form the right triangle. You can calculate the volume if any one of those dimensions is given, such as the side length of the cone.

Things You'll Need

Scientific calculator

Instructions

- 1
  Calculate the sine of the angle on a scientific calculator. For example, if the angle is 60 degrees, then sin(60) = 0.866.
- 2
  Multiply the side length of the cone by the sine of the angle to calculate the radius of the cone. For example, if the side length is 10 inches, then the radius is 10 x 0.866 = 8.66 inches.
- 3
  Calculate the cosine of the angle on the calculator. In this example, cos(60) = 0.5.
- 4
  Multiply the side length of the cone by the cosine of the angle to calculate the height of the cone. In this example, the height is 10 x 0.5 = 5 inches.
- 5
  Raise the radius of the cone to square. In this example, the square radius is 8.66 x 8.66 = 75.
- 6
  Multiply the square radius by the height and then by the constant "pi" (3.142); then divide the product by 3 to calculate the volume of the cone. In this example, the volume is (75 x 5 x 3.142) / 3 = 392.75 cubic inches.

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