Height of a Cone Calculator
Calculate the perpendicular height of a right circular cone from its base radius plus slant height (h = √(l²−r²)), or from its base radius plus volume (h = 3V/πr²).
Known inputs
units
units
h = √(l² − r²)
- 1
l² − r²
5² − 3² = 25 − 9 = 16Pythagoras: slant height is the hypotenuse, radius is one leg. - 2
Perpendicular height
√(16) = 4
4 units
How does this calculator work?
Cone height h = √(l² − r²) from slant height l and base radius r, or h = 3V/(π·r²) from volume V and radius r. Enter these two inputs to get height plus all derived cone dimensions: volume, slant height, base area, lateral surface area and total surface area.
Formula
How this is calculated
A right circular cone has a flat circular base of radius r and a single apex directly above the centre. The perpendicular height h is the vertical distance from the apex to the base centre.
If you know the slant height l — the straight-line distance from the apex to any point on the base rim — Pythagoras gives h = √(l² − r²). The three lengths h, r and l form a right triangle with l as the hypotenuse. For a real positive height, the slant height must strictly exceed the radius; if l ≤ r the geometry is impossible.
If you know the volume V instead, rearrange the cone volume formula V = (1/3)·π·r²·h to get h = 3V/(π·r²). All remaining dimensions — slant height, base area, lateral surface area (π·r·l) and total surface area — are derived and displayed automatically so you have a complete set of cone dimensions from just two inputs.
Frequently asked questions
Use Pythagoras: h = √(l² − r²). The slant height l, perpendicular height h and base radius r form a right triangle with l as the hypotenuse. The slant height must be greater than the radius for the solution to be real.
The perpendicular (true) height h is the vertical distance from the apex to the base centre. The slant height l is the distance from the apex to any point on the base rim. They relate by l = √(h² + r²), so l is always larger than h.
Yes. Total surface area = π·r·(l + r). Solve for l: l = SA/(π·r) − r. Then apply h = √(l² − r²). Run the slant-height mode after computing l from that rearrangement.
Also known as
TG we-Calculate Editorial Team. (2026). Height of a Cone Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/height-of-cone-calculator
TG we-Calculate Editorial Team. "Height of a Cone Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/height-of-cone-calculator.
TG we-Calculate Editorial Team, "Height of a Cone Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/height-of-cone-calculator
@misc{wecalculate_height_of_cone_calculator, title = {Height of a Cone Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/height-of-cone-calculator}}, year = {2026}, note = {TG we-Calculate} }
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