Pyramid Volume: Formula, Examples & Calculator

This calculator works out the volume of any pyramid from its base area and height, or from its base dimensions if the base is a square, rectangle, or triangle. You enter the base measurements and the perpendicular height, and you get the volume in cubic units (cubic meters, cubic feet, etc.) instantly.

Formula

The volume of a pyramid is always one-third of the volume of a prism with the same base and height. In symbols:

V = (1/3) × B × h

• V = volume (cubic units, e.g., m³, ft³, cm³)
• B = area of the base (square units, e.g., m², ft²)
• h = perpendicular height from the apex to the base (linear units, e.g., m, ft)

The base area depends on the shape of the base. For a square base, B = side². For a rectangular base, B = length × width. For a triangular base, B = (1/2) × base of triangle × height of triangle. For any other regular polygon, use its standard area formula.

Worked Example

Using the default values from the calculator above: a rectangular pyramid with base length 6 m, base width 4 m, and perpendicular height 10 m.

1. Calculate the base area: B = 6 × 4 = 24 m²
2. Multiply by the height: 24 × 10 = 240 m³
3. Divide by 3: 240 / 3 = 80 m³

The volume is 80 cubic meters. This matches the calculator’s output (rounded to 2 decimal places).

Why the One-Third Factor?

Imagine a cube divided into three identical pyramids. Each has a square base equal to a face of the cube and a height equal to the cube’s edge. Since the three pyramids fill the cube exactly, each pyramid’s volume is one-third of the cube’s volume. This geometric relationship holds for any pyramid, not just square-based ones, as proven by Cavalieri’s principle: if two solids have the same height and equal cross-sectional areas at every level, they have equal volumes.

How to Measure the Height Correctly

The height in the formula is the perpendicular distance from the apex (the top point) straight down to the plane of the base. It is not the slant height (the distance along a triangular face) and not the edge length from apex to a base corner. If you only know the slant height and the base dimensions, you must use the Pythagorean theorem to find the perpendicular height first. For a square pyramid with base side a and slant height l, the perpendicular height is √(l² – (a/2)²).

Common Mistakes and How to Avoid Them

• Using slant height instead of perpendicular height – always check that the height is measured at a right angle to the base.
• Mixing units – if the base is in meters and the height in centimeters, convert one to the other first. The volume will be in cubic units of the consistent unit.
• Forgetting to divide by 3 – the most common error. The factor of 1/3 is essential.
• Confusing base area with base side length – for a square base, you must square the side length before multiplying by height.
• Rounding too early – keep full precision until the final step, then round to the desired number of decimal places.

When to Use This Calculation

Pyramid volume calculations appear in architecture (designing roofs, monuments, or tents), construction (estimating concrete for a pyramidal foundation), packaging (pyramid-shaped containers), and even in math homework. The formula applies to any pyramid, including tetrahedrons (triangular base), square pyramids (like the Egyptian ones), and pyramids with pentagonal or hexagonal bases. For a cone, which is a pyramid with a circular base, the same formula works using the circle’s area (Ï€r²) as the base.

Limitations and Real-World Considerations

The formula assumes a perfect mathematical pyramid with straight edges and a flat base. Real objects may have slightly curved surfaces or irregular bases, so the result is an approximation. For a truncated pyramid (a pyramid with the top cut off), you need a different formula that subtracts the volume of the removed top. Also, if the base is not a regular polygon, you must calculate its area correctly, possibly by dividing it into triangles.

Frequently Asked Questions

What is the formula for the volume of a square pyramid?

For a square pyramid with base side length a and perpendicular height h, the volume is V = (1/3) × a² × h. For example, if the base side is 5 cm and the height is 9 cm, the volume is (1/3) × 25 × 9 = 75 cm³.

How do I find the volume of a triangular pyramid (tetrahedron)?

Use the same general formula: V = (1/3) × B × h, where B is the area of the triangular base. If the base is an equilateral triangle with side a, its area is (√3/4) × a². Then multiply by the pyramid’s height and divide by 3.

What is the difference between a pyramid and a prism?

A prism has two identical, parallel bases and rectangular side faces, so its volume is base area × height. A pyramid has one base and triangular side faces that meet at a single apex, so its volume is one-third of that: (1/3) × base area × height.

Does the volume formula work for a cone?

Yes. A cone is a pyramid with a circular base. The base area is πr², so the volume is (1/3) × πr² × h. This is the standard cone volume formula.

Why do I get a different answer if I use the slant height?

Because the slant height is longer than the perpendicular height. The formula requires the perpendicular height, which is the shortest distance from the apex to the base. Using the slant height will give a larger, incorrect volume.

Can I use this calculator for a pyramid with a pentagonal base?

Yes, if you know the base area. The calculator above only takes length and width, which suits rectangular bases. For a pentagonal base, compute its area separately (using its apothem and perimeter) and then multiply by the height and divide by 3.

Related

• How to Minus a Percentage on a Calculator
• File Is Corrupted and Cannot Be Opened? Fixes That Work
• Bose Speakers for Travel, Everyday Use & Home

in b | Base length | m | 6 | 0
in w | Base width | m | 4 | 0
in h | Height (perpendicular) | m | 10 | 0
out vol | Volume | m³ | (b * w * h) / 3 | 2
out base_area | Base area | m² | b * w | 2