Right Square Pyramid Calculator — Volume & Surface Area
Enter the base side length and perpendicular height of a right square pyramid to find its volume, slant height, lateral faces, total surface area and lateral edge.
units
units
V = ⅓ × a² × h
- 1
Base area
a² = 6 × 6 = 36 - 2
Volume
⅓ × 36 × 8 = 96V = ⅓ × base area × height — the universal pyramid formula.
How does this calculator work?
For a right square pyramid with base side a and height h: slant height l = √(h²+(a/2)²), Volume = ⅓a²h, Lateral SA = 2al, Total SA = a²+2al, Lateral edge = √(h²+a²/2). Enter base side and height — volume in unit³, areas in unit².
Formula
How this is calculated
A right square pyramid has a square base of side a and an apex directly above the centre of that base at perpendicular height h. Its four triangular lateral faces are congruent isosceles triangles.
Volume follows the universal pyramid formula: V = ⅓ × base area × height = ⅓a²h. The slant height l is the perpendicular distance from the apex to the midpoint of any base edge, forming a right triangle with legs h (the height) and a/2 (half the base side): l = √(h² + (a/2)²). Each triangular face has base a and height l, so its area is ½al; four faces give lateral SA = 2al. Adding the square base a² yields the total surface area a² + 2al.
The lateral edge connects the apex to a corner of the base. The horizontal distance from the centre to a corner is a√2/2, so the lateral edge is √(h² + a²/2). This is longer than the slant height. The apex angle is the angle at the apex of each triangular face, computed from the half-base and the slant height. All formulas assume the apex is exactly above the centre and all inputs are positive.
Frequently asked questions
The slant height l is the distance from the apex to the midpoint of a base edge, measured along the face: l = √(h² + (a/2)²). The lateral edge e is the distance from the apex to a corner of the base: e = √(h² + a²/2). The lateral edge is always longer.
The factor of ⅓ is a fundamental result: three identical pyramids — same square base and height — fit exactly inside a prism of the same dimensions. This is proved by integration of the cross-sectional area, which decreases quadratically from base to apex.
Yes. The Great Pyramid has an original base side of about 230.4 m and a height of about 146.5 m (current height 138.5 m). Entering those values gives the original volume (≈ 2.59 million m³) and total surface area. The result is exact for a perfect right square pyramid; the real pyramid's stepped sides introduce small deviations.
Also known as
TG we-Calculate Editorial Team. (2026). Right Square Pyramid Calculator — Volume & Surface Area [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/right-square-pyramid-calculator
TG we-Calculate Editorial Team. "Right Square Pyramid Calculator — Volume & Surface Area." TG we-Calculate. 2026. https://we-calculate.com/calculator/right-square-pyramid-calculator.
TG we-Calculate Editorial Team, "Right Square Pyramid Calculator — Volume & Surface Area," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/right-square-pyramid-calculator
@misc{wecalculate_right_square_pyramid_calculator, title = {Right Square Pyramid Calculator — Volume & Surface Area}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/right-square-pyramid-calculator}}, year = {2026}, note = {TG we-Calculate} }
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