Slant Height of a Cone Calculator
Enter the height and base radius of a right circular cone to find its slant height — the distance along the surface from any point on the base circumference to the apex. Also computes lateral area, total surface area, volume and the apex half-angle.
Distance along the curved surface from any base-circle point to the apex
- 1
h² (height squared)
12² = 144 - 2
r² (radius squared)
5² = 25 - 3
h² + r²
144 + 25 = 169Pythagorean theorem: slant height l is the hypotenuse of the right triangle formed by h and r. - 4
Slant height l = √(h² + r²)
√169 = 13
How does this calculator work?
Slant height l = √(h² + r²) for a cone with perpendicular height h and base radius r. Lateral area = πrl, total surface area = πr(r + l), volume = ⅓πr²h, apex half-angle = arctan(r/h). Slant height is always larger than perpendicular height and equals the hypotenuse of the right triangle formed by h and r.
Formula
How this is calculated
A right circular cone has its apex directly above the centre of its circular base. The slant height l is the straight-line distance from the apex to any point on the base circumference, running along the sloped surface. The perpendicular height h (apex to base centre) and the base radius r form a right angle at the base, so by the Pythagorean theorem l = √(h² + r²).
Slant height is essential for computing the lateral surface area. If you unroll the cone's curved surface it becomes a flat sector with radius l and arc length equal to the base circumference 2πr — so the lateral area = ½ × l × 2πr = πrl. Adding the circular base area πr² gives total surface area πr(r + l). Volume uses the universal cone/pyramid formula ⅓ × base area × height = ⅓πr²h.
The apex half-angle θ = arctan(r/h) gives the angular spread of the cone from its axis — useful in optics (spotlight beam angle), acoustics, and machining (drill point angle). All formulas assume a right circular cone (apex directly above base centre); oblique cones require more involved geometry.
Frequently asked questions
For a cone the lateral surface is smooth and every line from the apex to the base circumference has the same length — that common length is the slant height. For a pyramid (with flat triangular faces), the slant height is the height of a triangular face, while a lateral edge is the ridge between two adjacent faces; those two measurements differ.
Rearrange the formula: h = √(l² − r²). Enter l as your "height" input and r as the radius, then mentally swap the output labels. Alternatively just compute h = √(l² − r²) and re-enter the derived height alongside the radius.
Divide the diameter by 2 to get the radius, then apply l = √(h² + r²). The calculator uses radius as input because the Pythagorean formula uses r directly; if your measurement is diameter, simply halve it before entering.
Also known as
TG we-Calculate Editorial Team. (2026). Slant Height of a Cone Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/slant-height-of-cone-calculator
TG we-Calculate Editorial Team. "Slant Height of a Cone Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/slant-height-of-cone-calculator.
TG we-Calculate Editorial Team, "Slant Height of a Cone Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/slant-height-of-cone-calculator
@misc{wecalculate_slant_height_of_cone_calculator, title = {Slant Height of a Cone Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/slant-height-of-cone-calculator}}, year = {2026}, note = {TG we-Calculate} }
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