Pyramid Calculator — Surface Area & Volume

Regular pyramids (square, triangular, …). Private by design — runs locally in your browser.

Diagram & Inputs

Tip: Provide a and n, plus at least one of h, ℓ, S, or V. The calculator fills in the rest and flags inconsistencies.

Results

How the Pyramid Calculator Works

This tool solves any regular pyramid (base is a regular n-gon). Enter the base edge a and choose n. From there, give either the height h, the slant height ℓ, the total surface S, or the volume V. The calculator infers the rest, checks consistency within a small tolerance, and labels units for you.

  • Base perimeter: p = n·a
  • Base inradius: r = a / (2·tan(π/n))
  • Base area: B = n·a² / (4·tan(π/n))
  • Surface area: S = B + ½·p·ℓ
  • Volume: V = (1/3)·B·h
  • Relation: ℓ² = h² + r²
  • From S: ℓ = 2(S - B)/p (requires a and n)
  • From V: h = 3V/B (requires a and n)

Units: a, h, and ℓ use a length unit (e.g., cm); S uses the squared unit (e.g., cm²); V uses the cubed unit (e.g., cm³).

5 Fun Facts about Pyramids

Volume plays thirds

Any pyramid’s volume is ⅓ of a prism with the same base and height: V = (1/3)·B·h. Cones obey the same one‑third rule.

Prism sibling

Slant vs true height

Slant height ℓ and vertical height h meet via ℓ² = h² + r², where r is the inradius of the base polygon—Pythagoras hiding in the side face.

Pythagoras link

Square-base shortcut

For a square pyramid, each face is an isosceles triangle. Its altitude is √(ℓ² − (a/2)²), so total area splits neatly into four equal faces plus the base.

Face geometry

Egyptian slope

The Great Pyramid’s original slope was close to a 14:11 rise-run (~51.8°). Its height-to-base ratio almost matches a circle’s radius to half-circumference.

Historic curiosity

n-gon continuum

Change the base: triangle, pentagon, decagon—regular polygons all work. As n grows large, a regular pyramid approaches a cone.

Shape spectrum

Pyramid Calculator: FAQs

Which inputs are valid to solve a pyramid?

Provide a and n plus one of h, ℓ, S, or V. Two or more values are fine; the tool checks consistency.

What formulas are used?

B = n·a²/(4·tan(π/n)), S = B + ½·n·a·ℓ, V = (1/3)·B·h, and ℓ² = h² + r² with r = a/(2·tan(π/n)).

Does the calculator keep my data private?

Yes. Computation is entirely client-side; nothing is uploaded.

Can I change units or decimal places?

Yes. Choose a length unit for a, h, and ℓ. Surface area uses the squared unit; volume uses the cubed unit.

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