Cone Calculator for Volume and Surface Area
Find the volume, slant height, surface area and angles of a right circular cone
| Property | Value |
|---|
Cone calculator results cover every key dimension
Cone calculator results cover every key dimension including volume, slant height, lateral surface area, base area, total surface area and the half angle, aperture angle and base angle. The table below shows example results for a cone with a radius of 5 and a height of 12.
| Property | Formula | Example Value |
|---|---|---|
| Slant height (s) | √(r² + h²) | 13 |
| Volume (V) | (1/3)πr²h | 314.159 |
| Lateral area (L) | πrs | 204.204 |
| Base area (B) | πr² | 78.540 |
| Total area (A) | L + B | 282.743 |
Cone volume formula starts with one third pi r squared h
Cone volume formula starts with one third pi r squared h because a cone holds exactly one third the volume of a cylinder with the same base and height. The radius is squared, multiplied by pi and by the height, then divided by three to give the final volume in cubic units.
Finding slant height with the Pythagorean theorem
Finding slant height with the Pythagorean theorem treats the cone as a right triangle where the radius and height are the two legs and the slant height is the hypotenuse. The formula s equals the square root of r squared plus h squared applies directly from this right triangle relationship.
Lateral area versus total surface area
Lateral area versus total surface area describes two related but different measurements. Lateral surface area covers only the slanted outer surface, calculated as pi times radius times slant height. Total surface area adds the flat circular base, calculated as pi times radius times the sum of slant height and radius.
Calculating cone angles
Calculating cone angles starts with the half angle theta, found by taking the arctangent of radius divided by height. The aperture angle, also called the full angle at the apex, equals twice the half angle. The base angle equals 90 degrees minus the half angle since the radius and height meet at a right angle.
Reverse cone calculations solve for missing dimensions
Reverse cone calculations solve for missing dimensions when you know the volume, slant height or surface area instead of both the radius and height. For example, if you know the volume and radius, the height can be found by rearranging the volume formula to h equals 3 times V divided by pi r squared.
Solving for radius from height and volume
Solving for radius from height and volume uses the rearranged formula r equals the square root of 3 times V divided by pi times h. This is useful when designing a cone shaped container that must hold a specific volume within a fixed height.
Solving for height from slant height and radius
Solving for height from slant height and radius rearranges the Pythagorean relationship to h equals the square root of s squared minus r squared. This works whenever the slant height is the longest side of the right triangle formed by the cone cross section.
Frequently asked questions about cone calculations
The formula for the volume of a cone is V equals one third times pi times the radius squared times the height, written as V = (1/3)pi r^2 h. This gives the volume in cubic units based on the base radius and the vertical height.
The slant height of a cone is found using the Pythagorean theorem, s equals the square root of r squared plus h squared, where r is the base radius and h is the vertical height. The slant height represents the distance from the apex to the edge of the base along the surface.
Lateral surface area is the area of the curved side of the cone only, calculated as pi times radius times slant height. Total surface area adds the circular base area to the lateral area, calculated as pi times radius times the quantity slant height plus radius.
To calculate the height of a cone from its volume, rearrange the volume formula to h equals 3 times V divided by pi times r squared. This requires knowing the volume and the base radius of the cone.
A right circular cone is a three dimensional shape with a circular base where the apex sits directly above the center of the base, forming a perpendicular axis. This is the most common type of cone used in geometry problems and everyday objects.
The half angle of a cone is found using theta equals the arctangent of radius divided by height, which gives the angle between the central axis and the slant side. The base angle equals 90 degrees minus the half angle, and the full aperture angle equals twice the half angle.