Volume Equation for Pyramid: Formula & Examples
The calculator for this page is switched off while it is checked. The formula is below, and the worked example in the article shows how to use it by hand.
This calculator works out the volume of any pyramid from its base area and height, or directly from base dimensions for common shapes. You enter the base dimensions (or area) and the perpendicular height, and you get the volume in cubic units. The formula is always one-third of base area times height.
Formula
The volume V of a pyramid is given by:
V = (1/3) × B × h
- V = volume (cubic units)
- B = area of the base (square units)
- h = perpendicular height from the apex to the base (linear units)
For a rectangular base, B = length × width, so the formula becomes V = (l × w × h) / 3. The height must be measured perpendicular to the base, not along a slanted edge.
Worked Example
Using the default values from the calculator: base length = 6 m, base width = 4 m, height = 9 m.
- Compute the base area: B = 6 × 4 = 24 m².
- Multiply by the height: 24 × 9 = 216 m³.
- Divide by 3: 216 / 3 = 72 m³.
The calculator returns 72.00 m³, matching the manual calculation.
Why the 1/3 Factor?
The 1/3 factor comes from calculus: if you slice a pyramid horizontally, each slice is a scaled copy of the base. The area of a slice at height y is B × (y/h)², and integrating that from 0 to h gives B × h / 3. This is the same reason a cone (which is a pyramid with a circular base) uses the same formula. A pyramid fits exactly inside a prism with the same base and height, but it occupies one-third of the prism’s volume.
When to Use This Formula
Use this formula for any pyramid with a polygonal base: square, rectangular, triangular, pentagonal, etc. The shape of the base does not matter as long as you know its area. The apex can be directly above the center (a right pyramid) or off-center (oblique pyramid); the formula still works because the volume depends only on base area and perpendicular height.
Common applications include:
- Estimating the volume of a tent or roof structure
- Calculating the capacity of a hopper or pile of sand
- Solving geometry homework problems
- Determining the volume of a pyramid-shaped container
Common Mistakes and How to Avoid Them
The most frequent error is using the slant height instead of the perpendicular height. The slant height is measured along a face, not vertically from the apex to the base. Always use the height that is at a right angle to the base.
Another common mistake is forgetting to divide by 3. It is easy to compute base area × height and forget the factor. Always check: the volume of a pyramid is always less than the volume of a prism with the same base and height.
Unit mix-ups are also common. If you measure the base in meters and the height in centimeters, you must convert to the same unit before calculating. The volume will be in cubic units of whatever unit you use.
Limits of the Formula
The formula assumes the base is a flat polygon and the apex is a single point. It does not apply to a frustum (a pyramid with the top cut off) – that requires a different formula. It also does not apply to curved shapes like spheres or cylinders. For a cone, the same formula works because a cone is a pyramid with a circular base, but the base area is πr².
Frequently Asked Questions
Does the formula work for a triangular base?
Yes. As long as you calculate the area of the triangular base correctly (e.g., ½ × base × height of the triangle), the volume is still (1/3) × base area × pyramid height.
What is the difference between a pyramid and a prism?
A prism has two parallel, congruent bases and rectangular sides. A pyramid has one base and triangular sides that meet at a single apex. The volume of a prism is base area × height, while a pyramid is one-third of that.
Can I use the slant height instead of the height?
No. The slant height is longer than the perpendicular height and would give an incorrect volume. You must use the perpendicular distance from the apex to the base.
How do I find the base area of a pentagonal pyramid?
For a regular pentagon, the area is (1/2) × perimeter × apothem, or use the formula (5/4) × s² × cot(π/5) where s is the side length. Then apply the pyramid volume formula.
What if my pyramid is oblique?
The formula still works. The volume of an oblique pyramid is the same as a right pyramid with the same base and height, because the cross-sections parallel to the base have the same area distribution.
Related
Formula used
in b | Base length | m | 6 | 0 in w | Base width | m | 4 | 0 in h | Height | m | 9 | 0 out V | Volume | m³ | (b * w * h) / 3 | 2 out base_area | Base area | m² | b * w | 2
Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work
Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.