Height of a Square Pyramid Calculator
Calculate the perpendicular height of a square pyramid from its base side length plus slant height (h = √(l²−(a/2)²)), or from its base side length plus volume (h = 3V/a²).
Known inputs
units
units
h = √(l² − (a/2)²)
- 1
Half the base side
a ÷ 2 = 6 ÷ 2 = 3 - 2
l² − (a/2)²
5² − 3² = 25 − 9 = 16Pythagoras: the hypotenuse is the slant height, one leg is a/2. - 3
Perpendicular height
√(16) = 4
4 units
How does this calculator work?
Perpendicular height of a square pyramid: h = √(l² − (a/2)²) from slant height l and base side a, or h = 3V/a² from volume V. Volume is V = a²h/3. Slant height l = √(h² + (a/2)²); lateral edge = √(h² + a²/2).
Formula
How this is calculated
A square pyramid has a square base with side length a and four identical triangular faces that meet at a single apex. The perpendicular height h is the straight-line distance from the apex directly down to the centre of the base — the altitude of the solid.
If you know the slant height l — the distance from the apex to the midpoint of one base edge — recover h using Pythagoras. The right triangle formed by h (vertical leg), a/2 (horizontal leg from base centre to edge midpoint) and l (hypotenuse) gives h = √(l² − (a/2)²). For a real solution, the slant height must be strictly greater than a/2; if l ≤ a/2 the geometry is impossible.
If you know the volume V instead, rearrange the square-pyramid volume formula V = (a²·h)/3 to get h = 3V/a². All remaining dimensions — derived slant height, lateral edge (apex to corner), base area, lateral and total surface area — are shown in the secondary grid so you get a complete picture from a single calculation.
Frequently asked questions
The slant height l is the distance from the apex to the midpoint of any base edge — also the height of one triangular face. It is always longer than the perpendicular height h, and they relate by l = √(h² + (a/2)²).
There are four identical triangular faces, each with base a and height equal to the slant height l. Each has area (1/2)·a·l, so the total lateral surface area is 4 × (1/2)·a·l = 2·a·l.
The lateral edge is the straight line from the apex to one corner of the base. For base side a and perpendicular height h it equals √(h² + (a·√2/2)²) = √(h² + a²/2).
Also known as
TG we-Calculate Editorial Team. (2026). Height of a Square Pyramid Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/height-of-a-square-pyramid-calculator
TG we-Calculate Editorial Team. "Height of a Square Pyramid Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/height-of-a-square-pyramid-calculator.
TG we-Calculate Editorial Team, "Height of a Square Pyramid Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/height-of-a-square-pyramid-calculator
@misc{wecalculate_height_of_a_square_pyramid_calculator, title = {Height of a Square Pyramid Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/height-of-a-square-pyramid-calculator}}, year = {2026}, note = {TG we-Calculate} }
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