Volume of a Pyramid Calculator

Calculate the volume of square, rectangular, and triangular pyramids: V = ⅓ × Base Area × Height. Find lateral surface area and slant height for any pyramid type.

📐 Geometry📐 V = ⅓ × B × h🔢 Math
Pyramid type
Base side s (or length l)
Width w (rectangular only)
Height h
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Volume of a Pyramid Calculator—V = ⅓ × B × hcubic units

Step-by-Step Examples

Example 1
Square Pyramid

s=4, h=6.

  • B=16, V=⅓×16×6=32
✓ V = 32 cubic units
Example 2
Great Pyramid

Base 230m × 230m, h=146.5m.

  • B=52,900 m²
  • V=⅓×52900×146.5≈2,582,700 m³
✓ V ≈ 2,583,000 m³
Example 3
Triangular

Equilateral base s=3, h=5.

  • B=√3/4×9≈3.897
  • V=⅓×3.897×5≈6.495
✓ V ≈ 6.50 cubic units

Real-World Applications

🏛️
Architecture
Egyptian pyramids, modern glass pyramid structures.
📐
Geometry
Pyramid is the 3D analog of a triangle — base tapers to a point.
🧪
Engineering
Roof structures, hoppers, and silos use pyramid geometry.
📊
Calculus
Pyramid volume is derived by integration — slices of decreasing squares.

Common Mistakes to Avoid

⚠️
V = ⅓ not ½

Pyramid volume is ⅓ × base × height. Not ½. (½ is for triangle area, not pyramid volume.)

⚠️
Height is perpendicular

h is the perpendicular distance from base to apex, not the slant edge length.

⚠️
Slant height vs lateral edge

These are different! Slant height: from midpoint of base edge to apex. Lateral edge: from corner of base to apex.

Frequently Asked Questions

Why ⅓ in the formula? ▾
Three congruent pyramids can fill a cube. Proof by Cavalieri's principle or integration.
What is a regular pyramid? ▾
A pyramid with a regular polygon base and apex directly above the center of the base.
What is the lateral surface area? ▾
Sum of all triangular faces. For square pyramid: 4×(½×s×l) = 2sl where l is slant height = √(h²+(s/2)²).
What is a tetrahedron? ▾
A triangular pyramid where all four faces are equilateral triangles. A regular tetrahedron with edge a has V=a³/(6√2).
How is pyramid related to prism? ▾
A prism with same base and height has volume = B×h. Pyramid has ⅓ of that volume.

Related Math Calculators

📐 Cone Volume📐 Volume📐 Parallelogram📐 Surface Area📐 Heron Formula

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

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