Pyramid Calculator
Find the volume and surface area of a rectangular or square-base pyramid from its base length, width, and height.
🔺 What is the Pyramid Calculator?
This pyramid calculator finds the volume and total surface area of a right rectangular or square-base pyramid using V=(1/3)LWh. Enter the base length, width, and height, and it returns the volume, total surface area, lateral (side) area, and base area.
For a square base with a clean 3-4-5 right-triangle slant height (length=width=6, height=4), this calculator gives an exact volume of 48 and total surface area of 96.
A pyramid always has exactly one-third the volume of a prism sharing the same base and height, a general geometric fact this calculator applies directly.
This calculator is useful for geometry students, and for anyone estimating the volume or material needed for a pyramidal structure, roof, or hopper.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Square base with exact 3-4-5 slant
Example 2 - Rectangular base
Example 3 - Great Pyramid of Giza scale
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for pyramid volume?
V = (1/3)LWh, where L and W are the base length and width, and h is the perpendicular height from the base to the apex. This applies to any right rectangular or square-base pyramid.
What is the formula for pyramid surface area?
Total surface area = base area (L×W) plus the lateral area of the four triangular faces. The two faces along the length have slant height s₁=√(h²+(W/2)²), and the two faces along the width have slant height s₂=√(h²+(L/2)²), giving lateral area = L×s₁ + W×s₂.
Why is a pyramid's volume exactly one-third of a prism with the same base and height?
This is a general geometric fact (provable with calculus or Cavalieri's principle) that applies to any pyramid or cone: it always occupies exactly one-third the volume of the prism or cylinder sharing the same base area and height, regardless of the base's shape.
What is slant height, and how is it different from the pyramid's height?
The pyramid's height is the straight vertical distance from the base to the apex. The slant height is the distance from the apex down the middle of a triangular face to the midpoint of a base edge, always longer than the vertical height (unless the base edge has zero length).
What is a real example with known dimensions?
The Great Pyramid of Giza has a square base of about 230.4 m and an original height of about 146.6 m, giving a volume of roughly 2.6 million cubic metres using this exact formula, closely matching published estimates.
Does this calculator work for a square-base pyramid?
Yes, a square base is simply the special case where length equals width (L=W), just enter the same value for both fields and the formulas automatically simplify to the standard square-pyramid case.
What if the pyramid's apex isn't centered over the base?
This calculator assumes a right pyramid, where the apex sits directly above the center of the base, the standard case for most textbook problems and real pyramidal structures. An oblique pyramid (off-center apex) has the same volume formula but a more complex surface area calculation not covered here.
How does pyramid volume compare to a cone's volume?
A cone is essentially a pyramid with a circular base, and its volume formula V=(1/3)πr²h follows the exact same one-third-of-a-prism (or cylinder) pattern, just with a circular base area (πr²) instead of a rectangular one (LW).
What units does this calculator use?
Any consistent length unit works, if you enter length, width, and height in metres, the volume comes out in cubic metres and the surface area in square metres, the formulas are unit-agnostic as long as all three inputs use the same unit.
Can I use this for a triangular-base pyramid (tetrahedron)?
No, this calculator is specifically for rectangular and square-base pyramids. A triangular-base pyramid (tetrahedron) uses a different base-area formula and different slant-height geometry for its three triangular faces.