Math
Cone Calculator
The cone calculator solves a right circular cone from any sensible pair — radius and height, radius and slant height, height and slant height, diameter and height, or radius and volume.
It returns the volume (exact in π), the slant height from the Pythagorean theorem, lateral and total surface area, and the apex and base angles.
Radius, height, or diameter depending on the mode.
Height, slant height, or volume (cubic units).
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Volume
—
- Volume (exact)
- —
- Radius, height, slant height
- —
- Lateral (curved) area
- —
- Total surface area
- —
- Base area
- —
- Apex angle / base angle
- —
More details (1 more)
- Capacity
- —
Student quick launch
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Study path
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What the cone calculator solves
The cone calculator solves a right circular cone from any sensible pair — radius and height, radius and slant height, height and slant height, diameter and height, or radius and volume. It returns the volume (exact in π), the slant height from the Pythagorean theorem, lateral and total surface area, and the apex and base angles.
Formula
How to use it
- Pick the pair of measurements you have. Slant height is the distance along the sloped surface from the tip to the base edge.
- Enter the two values in one unit and select it.
- Read the volume, slant height, surface areas, and angles; open the work for the Pythagorean step.
How to read the answer
A cone holds exactly one third of the cylinder with the same base and height. The slant height, height, and radius form a right triangle (l is the The longest side of a right triangle, always opposite the right angle.), which is why the slant must be longer than both. The lateral area is the curved surface unrolled into a sector of radius l; the total area adds the circular base.
Common mistakes and edge cases
- Using the slant height in the volume formula — volume needs the vertical height h.
- Forgetting the 1/3: a cone is one third of the matching cylinder.
- A slant height that is not longer than the radius (or the height) is impossible; the calculator rejects it.
- Entering the diameter as the radius; use the diameter mode instead.
Worked examples
Radius 3, height 4 (12π, slant 5)
Volume
37.6991
Radius 3, slant height 5
Volume
37.6991
Height 12, slant 13 (radius 5)
Volume
314.1593
Diameter 10, height 12
Volume
314.1593
Radius 3, volume 50 → height
Volume
50
Decimal radius 2.5, height 6
Volume
39.2699
Slant shorter than radius (impossible)
Volume
Error
Frequently asked questions
What is the volume of a cone?+
V = (1/3)πr²h — one third of the cylinder with the same radius and height. Radius 3 and height 4 give 12π ≈ 37.70.
How do I find the slant height of a cone?+
Pythagorean theorem with the radius and height: l = sqrt(r² + h²). Radius 3, height 4 gives slant 5.
What is the surface area of a cone?+
Lateral (curved) area is πrl; total surface area adds the base: πrl + πr². With r = 3 and l = 5 the total is 24π.
How do I find the height of a cone from the volume?+
h = 3V / (πr²). A cone of volume 50 with radius 3 is about 5.305 tall.
What is the apex angle of a cone?+
The full angle at the tip, 2·arctan(r/h). A cone with r = 3, h = 4 has an apex angle of about 73.74°.
Does the calculator handle a truncated cone?+
Not here — use the frustum option on the volume calculator for a cone with the top cut off.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026