Pyramid Volume Calculator
This pyramid volume calculator handles right pyramids whose apex sits directly above the centre of the base. Example: a square base with a = 6 and height 4 has a volume of 48 and a surface area of 96.
Example: Base shape: Square · Base side a 6 · Pyramid height h 4 → Result: Volume V = 48 · surface area S = 96 (base area B = 36). Source: Wikipedia – Pyramid (geometry). Updated: .
How it is calculated
A pyramid has a base and lateral faces that meet in a common apex. Its volume is always one third of base area times height.
Formulas and steps
The volume is V = B · h / 3 and the surface area S = B + L. For the lateral area L you need the slant height s, the height of a triangular face. For a square base s = √(h² + (a/2)²) and L = 2 · a · s. A rectangle has two slant heights, √(h² + (b/2)²) over side a and √(h² + (a/2)²) over side b. With an equilateral triangle as base, B = a² · √3 / 4 and s = √(h² + ρ²) where ρ = a · √3 / 6.
Worked examples
Square with a = 6 and h = 4: B = 36, V = 36 · 4 / 3 = 48. The slant height is √(16 + 9) = 5, the lateral area 2 · 6 · 5 = 60 and the surface area 36 + 60 = 96. Rectangle 6 × 8 with h = 4: V = 64, lateral area ≈ 73.9411, surface area ≈ 121.941. Triangle with a = 6 and h = 4: V ≈ 20.7846 and S ≈ 54.8185.
- The slant height is always longer than the pyramid height because it is the hypotenuse of a right triangle.
- Doubling the height at the same base doubles the volume; doubling the base side length quadruples it.
Frequently asked questions
What is the formula for the volume of a pyramid?
V = base area · height / 3. With B = 36 and h = 4 you get V = 48.
How do I find the surface area?
Base area plus lateral area. The lateral area consists of the triangular faces and needs the slant height, 5 in the example.
What is the difference between height and slant height?
The height is measured perpendicular from the apex to the base, the slant height from the apex to the midpoint of a base edge. It is longer, 5 against 4 in the example.
Sources and legal basis
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