Cone Diameter Calculator - Calculate from Height & Radius

What is the diameter of a cone?

The diameter of a cone is the width of its circular base, measured in a straight line through the center. It equals twice the base radius. A cone can also be described by its height (the straight distance from the base to the apex) and its slant height (the distance from the apex down the outside of the cone to the edge of the base). If any two of these three values are known, the third can be found.

Cone diameter formula

There are two common ways to find the diameter, depending on which measurements are available.

From height and slant height:

$d = 2\sqrt{s^2 - h^2}$

Here d is the diameter, s is the slant height, and h is the height. This comes from the Pythagorean theorem: the slant height is the hypotenuse of a right triangle formed by the height and the radius, so $s^2 = h^2 + r^2$. Solving for the radius and doubling it gives the diameter.

From radius:

$d = 2r$

This is just the definition of diameter. If the radius is already known, no other measurement is needed.

The slant height must always be longer than the height, because it is the hypotenuse of a right triangle and a hypotenuse is always the longest side. If the slant height is equal to or shorter than the height, the shape described is not a valid cone.

How to calculate the diameter of a cone

Method 1: height and slant height

1. Measure the height (h), the straight vertical distance from the base to the apex.
2. Measure the slant height (s), the distance from the apex to the edge of the base along the cone's surface.
3. Square both values, subtract the height squared from the slant height squared, and take the square root: $\sqrt{s^2 - h^2}$.
4. Multiply the result by 2 to get the diameter.

Method 2: radius

1. Measure the radius (r), the distance from the center of the base to its edge.
2. Multiply by 2.

Only one of these two methods should be used at a time. Providing height, slant height, and radius together is ambiguous, since the three values might not describe a consistent cone.

Worked examples

Example 1: height and slant height

A cone has a height of 3 units and a slant height of 5 units.

$d = 2\sqrt{5^2 - 3^2} = 2\sqrt{25 - 9} = 2\sqrt{16} = 2 \times 4 = 8$

The diameter is 8 units.

Example 2: radius

A cone has a radius of 4 units.

$d = 2 \times 4 = 8$

The diameter is 8 units. This matches Example 1, since a cone with height 3 and slant height 5 has a base radius of 4.

Example 3: a flat cone

A cone has a height of 0.1 units and a slant height of 10 units.

$d = 2\sqrt{10^2 - 0.1^2} = 2\sqrt{100 - 0.01} = 2\sqrt{99.99} \approx 19.999$

The diameter is about 20.00 units. When the height is very small compared to the slant height, the diameter approaches twice the slant height.

Example 4: a needle-like cone

A cone has a height of 9.99 units and a slant height of 10 units.

$d = 2\sqrt{10^2 - 9.99^2} = 2\sqrt{100 - 99.8001} = 2\sqrt{0.1999} \approx 0.894$

The diameter is about 0.89 units. When the slant height is only slightly longer than the height, the cone is tall and thin, and its base diameter is small.

Where cone diameter calculations are used

Cone diameter shows up wherever conical shapes are measured or built. Engineers use it when designing funnels, nozzles, and conical tanks. Manufacturers use it to size molds and conical parts. Geologists use similar measurements to describe volcanic cones. The same relationship between height, slant height, and radius applies to any right circular cone, from a paper party hat to a rocket nose cone.

History

The mathematical study of cones goes back to ancient Greece. Apollonius of Perga, working around 200 BC, wrote a major treatise called Conics that examined how planes cut through cones to produce circles, ellipses, parabolas, and hyperbolas. The basic relationship between a cone's height, radius, and slant height, based on the Pythagorean theorem, has been known since antiquity and is still the method used today.

Frequently asked questions

What is the diameter of a cone?

It is the width of the cone's circular base, measured straight across through the center. It is twice the base radius.

How do you find the diameter of a cone without the radius?

Use the height and slant height with the formula $d = 2\sqrt{s^2 - h^2}$. This applies the Pythagorean theorem to find the radius, then doubles it.

What is the relationship between a cone's diameter, height, and slant height?

The height, the radius, and the slant height form a right triangle. The slant height is the hypotenuse, the height is one leg, and the radius is the other leg: $s^2 = h^2 + r^2$. The diameter is twice the radius.

Why does the calculator reject some values?

The height must be greater than zero. The slant height must be greater than the height, since a hypotenuse cannot be shorter than or equal to a leg. The radius must be greater than zero. Height and slant height, or radius, should be entered on their own; entering all three at once is ambiguous and is rejected.

Can a cone have two different diameters?

A standard cone has one diameter, at its base, and comes to a point at the apex. A frustum, which is a cone with the pointed top sliced off, has two diameters, one at each flat end.

What units should be used?

Any consistent unit works, such as centimeters, meters, or inches. The result comes out in the same unit used for the inputs.

References

1. Weisstein, Eric W. "Cone." MathWorld, Wolfram Research. https://mathworld.wolfram.com/Cone.html
2. O'Connor, J.J. and Robertson, E.F. "Conic sections." MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Conic_sections/