500
units³
7.937
units
500
units³
7.937
units
The Pyramid Volume Calculator provides a universal tool for computing the volume of any pyramid, regardless of its base shape. Whether your pyramid has a triangular, square, rectangular, pentagonal, hexagonal, or any other polygonal base, the fundamental volume formula remains the same:
$$V = \frac{1}{3} A h$$
where $$A$$ is the area of the base and $$h$$ is the perpendicular height from the base to the apex. This formula is one of the cornerstone results of solid geometry, applicable to both right pyramids (where the apex is directly above the centroid of the base) and oblique pyramids (where the apex is offset).
The one-third factor is what distinguishes a pyramid from a prism. If you filled a prism and a pyramid of the same base area and height with water, the prism would hold exactly three times as much. This was known empirically by ancient Egyptian builders and proved mathematically by Greek geometers using the method of exhaustion, a precursor to integral calculus.
In modern applications, the pyramid volume formula appears throughout engineering, earth sciences, and material estimation. Geologists estimate the volume of volcanic peaks modeled as cones or pyramids. Civil engineers calculate stockpile volumes for aggregates, sand, and grain stored in pyramidal heaps. Architects determine the internal volume of pyramid-shaped spaces for HVAC sizing. Material scientists compute the volume of pyramidal indentations in hardness testing (Vickers and Knoop indentation tests use pyramid-shaped diamond indenters).
This calculator also provides the equivalent cube side length — the edge length of a cube with the same volume as your pyramid. This metric offers an intuitive sense of scale: a pyramid with a volume of 1000 cubic units is equivalent to a cube with sides of 10 units. By entering just two values — base area and height — you get immediate, precise results for any pyramid configuration.
To use this calculator effectively, first determine your base area separately. For a square base with side $$a$$, the area is $$a^2$$. For a rectangle with sides $$a \times b$$, it is $$ab$$. For a triangle with base $$b$$ and height $$h_t$$, the area is $$\frac{1}{2}bh_t$$. For a regular hexagon with side $$s$$, the area is $$\frac{3\sqrt{3}}{2}s^2$$. Once you have the base area, enter it along with the perpendicular height to get the pyramid volume instantly.
Enter the base area $$A$$ (in square units) and the perpendicular height $$h$$. The calculator multiplies the base area by the height and divides by 3 to produce the volume. It also computes the cube root of the volume to give the equivalent cube side length for easy size comparison.
The volume output is the total three-dimensional space enclosed by the pyramid, expressed in cubic units consistent with your input units. The equivalent cube side is the edge length of a cube having the same volume, giving an intuitive sense of scale. Remember that this formula requires the perpendicular height (the shortest distance from the apex to the base plane), not a slant or edge measurement.
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Results
A pyramid with a 10×10 square base (area = 100) and height 15 has a volume of 500 cubic units, equivalent to a cube with side ~7.94.
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A pyramid with a triangular base of area 43.3 and height 20 has volume ~288.67 cubic units.
Yes. The formula $$V = \frac{1}{3}Ah$$ applies to any pyramid, regardless of the base shape (triangular, square, pentagonal, hexagonal, irregular) and regardless of whether it is a right or oblique pyramid. The only requirement is that $$h$$ is the perpendicular height from the base plane to the apex.
Calculate the base area first. For a square with side $$a$$: area = $$a^2$$. For a rectangle $$a \times b$$: area = $$ab$$. For a triangle with base $$b$$ and height $$h_t$$: area = $$\frac{1}{2}bh_t$$. For a regular polygon with $$n$$ sides of length $$s$$: area = $$\frac{ns^2}{4}\cot(\pi/n)$$.
This can be proven by dissection (cutting a prism into three pyramids of equal volume) or by calculus (integrating cross-sectional areas that decrease quadratically from base to apex). The 1/3 factor arises because the cross-sectional area at height $$y$$ scales as $$(1 - y/h)^2$$, and the integral of this from 0 to $$h$$ yields $$h/3$$.
For a right pyramid with a square base of side $$a$$ and slant height $$l$$ (apex to midpoint of base edge), use $$h = \sqrt{l^2 - (a/2)^2}$$. For a circular cone with radius $$r$$ and slant height $$l$$: $$h = \sqrt{l^2 - r^2}$$.
Yes. A cone is a pyramid with a circular base. Compute the base area as $$A = \pi r^2$$ and enter it along with the cone's height. The volume formula $$V = \frac{1}{3}\pi r^2 h$$ is a special case of the general pyramid volume formula.
It is the edge length of a cube having the same volume as your pyramid: side = $$\sqrt[3]{V}$$. This provides an intuitive comparison. For example, a pyramid volume of 8000 cubic units is equivalent to a cube with 20-unit sides.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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