Height of Cone Calculator - Calculate Cone Height Online
Find a cone's height from its radius and slant height with this free calculator based on the Pythagorean theorem. Includes formula, worked examples, and FAQ.
Height of Cone Calculator
Documentation
A cone height calculator finds the vertical height of a right circular cone from two measurements: the radius of its base and its slant height. It uses the Pythagorean theorem, since the height, the radius, and the slant height form a right triangle inside the cone.
What is the height of a cone?
The height of a cone is the straight-line distance from the apex (the pointed tip) down to the center of the circular base, measured at a right angle to the base. This is sometimes called the "vertical height" or "perpendicular height," to tell it apart from the slant height.
The slant height is different. It is the distance from the apex to a point on the edge of the base, measured along the outside surface of the cone. Because the slant height runs at an angle rather than straight down, it is always longer than the vertical height.
Cone height formula
For a right circular cone, the radius (r), the height (h), and the slant height (l) form a right triangle. The slant height is the hypotenuse. This means the height can be found with the Pythagorean theorem:
h = √(l² − r²)
Where:
- h is the height of the cone
- l is the slant height
- r is the radius of the base
The formula only works when the slant height is greater than the radius. If it is not, no real cone can exist with those measurements, because the hypotenuse of a right triangle must be longer than either of its other two sides.
How to calculate cone height
- Measure or note the radius of the cone's base.
- Measure or note the slant height, the distance along the outside of the cone from the tip to the edge of the base.
- Square both numbers, subtract the radius squared from the slant height squared, and take the square root of the result.
- Make sure both measurements use the same unit, such as centimeters or inches. The result will be in that same unit.
Worked examples
Example 1. A cone has a radius of 3 units and a slant height of 5 units.
h = √(5² − 3²) = √(25 − 9) = √16 = 4 units
Example 2. A cone has a radius of 5 units and a slant height of 13 units.
h = √(13² − 5²) = √(169 − 25) = √144 = 12 units
Example 3. A cone has a radius of 8 units and a slant height of 10 units.
h = √(10² − 8²) = √(100 − 64) = √36 = 6 units
When there is no valid height
If the slant height is equal to or smaller than the radius, the calculator cannot return a height. A cone cannot have a slant height equal to its radius, because that would require a height of zero or an imaginary number. For example, a radius of 5 and a slant height of 5 describes a flat disc, not a cone, so the calculator flags this input as invalid instead of showing a result.
Where cone height is used
Cone height, together with the radius, is used to calculate a cone's volume (V = ⅓πr²h) and its surface area. Architects use it when designing conical roofs. Manufacturers use it to work out how much material a conical part needs. It also comes up in geometry classes, since the cone is one of the basic three-dimensional shapes taught alongside the cylinder and the sphere.
Frequently asked questions
How do you find the height of a cone from its radius and slant height? Square the slant height, square the radius, subtract the second result from the first, then take the square root: h = √(l² − r²).
What is the formula for the height of a cone? h = √(l² − r²), where h is the height, l is the slant height, and r is the radius of the base. It comes directly from the Pythagorean theorem.
Can the height of a cone be greater than its slant height? No. The slant height is the hypotenuse of the right triangle formed by the height and the radius, and a hypotenuse is always the longest side of a right triangle.
What happens if the radius equals the slant height? There is no valid cone. A radius equal to (or greater than) the slant height would give a height of zero or less, which is not possible for a real cone.
How is slant height different from height? Height is measured straight down from the apex to the center of the base. Slant height is measured along the outer surface, from the apex to the edge of the base. Slant height is always the longer of the two.
What units does the result use? The result uses the same unit as the radius and slant height entered. If both are in centimeters, the height is given in centimeters.
References
- Weisstein, Eric W. "Cone." MathWorld, Wolfram Research. https://mathworld.wolfram.com/Cone.html
- "Cone (geometry)." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Cone_(geometry)