Slant Height Calculator — Square Pyramid
Enter the perpendicular height and base side length of a square pyramid to find its slant height — the distance along a triangular face from the midpoint of a base edge to the apex. Also computes lateral area, total surface area and volume.
Distance along a triangular face from base-edge midpoint to apex
- 1
Half the base side b ÷ 2
10 ÷ 2 = 5 - 2
h² + (b/2)²
12² + 5² = 169Pythagorean theorem: slant height is the hypotenuse of the right triangle formed by h and b/2. - 3
Slant height l = √(h² + (b/2)²)
√169 = 13
How does this calculator work?
Slant height l = √(h² + (b/2)²) for a square pyramid with perpendicular height h and base side b. Lateral area = 2bl, total surface area = b² + 2bl, volume = ⅓b²h. Slant height is always larger than perpendicular height and is the along-face distance from base-edge midpoint to apex.
Formula
How this is calculated
The slant height l of a square pyramid is the distance measured along a triangular face from the midpoint of a base edge up to the apex — not the same as the perpendicular height h, which drops straight from apex to base centre. These two segments, together with half the base side b/2, form a right triangle inside the pyramid: h is the vertical leg, b/2 is the horizontal leg, and l is the hypotenuse. Applying the Pythagorean theorem gives l = √(h² + (b/2)²).
This formula applies to a right square pyramid — a pyramid with a square base whose apex is directly above the centre of the base (the most common type in geometry problems). For rectangular or triangular bases the same principle applies but the "half base" dimension varies by which face you are measuring.
Slant height is needed for lateral surface area. Each of the four triangular faces has base b and height l (not h), giving area ½ × b × l per face and total lateral area 4 × ½ × b × l = 2bl. Adding the square base area b² gives the total surface area b² + 2bl. Volume follows the universal pyramid formula ⅓ × base area × height = ⅓ b² h.
Frequently asked questions
Perpendicular height h is the straight vertical distance from the apex to the centre of the base. Slant height l is the distance along the sloped triangular face from the midpoint of a base edge to the apex. Since l² = h² + (b/2)², slant height is always greater than perpendicular height. If you lay a ruler along a face of the pyramid, you measure slant height.
Yes. From total area A = b² + 2bl, solve for l: l = (A − b²) / (2b). Then recover h from h = √(l² − (b/2)²). Alternatively, if you know only lateral area = 2bl, simply l = lateral area / (2b).
The area of a triangle is ½ × base × (height of the triangle). The "height of the triangle" for each lateral face is the slant height l — it is perpendicular to the base edge of that face. The pyramid's perpendicular height h is perpendicular to the base plane, not to the base edge of the triangular face, so h cannot be used directly in the area formula.
Also known as
TG we-Calculate Editorial Team. (2026). Slant Height Calculator — Square Pyramid [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/slant-height-calculator
TG we-Calculate Editorial Team. "Slant Height Calculator — Square Pyramid." TG we-Calculate. 2026. https://we-calculate.com/calculator/slant-height-calculator.
TG we-Calculate Editorial Team, "Slant Height Calculator — Square Pyramid," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/slant-height-calculator
@misc{wecalculate_slant_height_calculator, title = {Slant Height Calculator — Square Pyramid}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/slant-height-calculator}}, year = {2026}, note = {TG we-Calculate} }
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