How to Solve the Circular Height for the Base of a Cone

Taller cones have larger volumes.

The height from a cone's circular base to its apex is proportional the the shape's total volume. This volume is also proportional to the base's surface area, which is, in turn, proportional to the base's radius. The entire cone has a volume equal to a third of the volume of a cylinder with a similar base. This means that a cone and a sphere based on a similarly sized circle together make up twice the volume of a cylinder based on that circle.

Instructions

1. Square the radius of the cone's base. If, for instance, it has a radius of 6 inches, then 6^2 or 6 x 6 equals 36.

2. Multiply this answer by Pi, which is approximately 3.142: 36 --- 3.142 equals 113.1 squared inches. This is the base's surface area.

3. Divide the cone's volume by this area. For instance, if it has a radius of 500 cubic inches: 500 divided by 113.1 equals 4.42.

4. Multiply this answer by 3: 4.42 --- 3 = 13.26. This is the cone's height, measured in inches.

References

• Andrews University: Lateral & Surface Areas, Volumes; Keith G. Calkin; March 2006
• North Carolina State University: What dx Actually Means; Kenny Felder; 1996

Resources

• University of Dayton: Archimedes of Syracuse
• The University of Tennessee: Volumes; Lawrence S. Husch; 2001
• Philadelphia University: Cones and Pyramids