Pyramid Volume: worked example
| Base length | 6 |
|---|---|
| Base width | 4 |
| Height | 9 |
| Volume | 72 |
| Total surface area | 117.3 |
Pyramid Volume: formula
Enter the length and width of the rectangular base and the vertical height from the base to the apex. The height is the perpendicular distance, not the slanted edge. The calculator gives volume, one third of the base area times the height, and total surface area, including the base and four triangular faces whose slant heights come from the Pythagorean theorem.
Use one unit for all measurements.
Pyramid Volume: changing Base length
| Base length | Volume | Total surface area |
|---|---|---|
| 4.8 | 57.6 | 100.7 |
| 5.4 | 64.8 | 109 |
| 6 (example) | 72 | 117.3 |
| 6.6 | 79.2 | 125.6 |
| 7.2 | 86.4 | 134 |
Pyramid Volume: compare input ranges
| Input | Tested from | To | Volume: low input | Volume: high input | Swing |
|---|---|---|---|---|---|
| Height | 8 | 10 | 64 | 80 | 16 (22%) |
| Base length | 5.4 | 6.6 | 64.8 | 79.2 | 14.4 (20%) |
| Base width | 3.6 | 4.4 | 64.8 | 79.2 | 14.4 (20%) |
Pyramid Volume: individual input comparisons
| Input | Value | Volume | Total surface area |
|---|---|---|---|
| Base length | 4.8 | 57.6 | 100.7 |
| 5.4 | 64.8 | 109 | |
| 6 | 72 | 117.3 | |
| 6.6 | 79.2 | 125.6 | |
| 7.2 | 86.4 | 134 | |
| Base width | 3.2 | 57.6 | 104.4 |
| 3.6 | 64.8 | 110.8 | |
| 4 | 72 | 117.3 | |
| 4.4 | 79.2 | 123.7 | |
| 4.8 | 86.4 | 130.2 | |
| Height | 7 | 56 | 98.1 |
| 8 | 64 | 107.7 | |
| 9 | 72 | 117.3 | |
| 10 | 80 | 126.9 | |
| 11 | 88 | 136.7 |
Pyramid Volume: combined input changes
| Base length / Base width | 3.2 | 4 | 4.8 |
|---|---|---|---|
| 4.8 | 46.1 | 57.6 | 69.1 |
| 5.4 | 51.8 | 64.8 | 77.8 |
| 6 | 57.6 | 72 | 86.4 |
| 6.6 | 63.4 | 79.2 | 95 |
| 7.2 | 69.1 | 86.4 | 103.7 |
Pyramid Volume: which assumptions move the result?
Pyramid Volume: values and definitions
| Quantity and definition | Value used |
|---|---|
| Base length | 6 |
| Base width | 4 |
| Height — The straight-up distance from the base to the tip, not the slanted edge. | 9 |
| Volume | 72 |
| Total surface area | 117.3 |
Pyramid Volume: half and double Height
| Height | Volume |
|---|---|
| 4.5 | 36 |
| 9 | 72 |
| 18 | 144 |
Common error: Forgetting the division by three. The result then matches a box, which is three times too large.
The formula assumes the apex is directly above the center of the base, a right pyramid. Truncated pyramids have a different formula. The slant heights differ for the two pairs of faces when the base is not square.
Unit errors are common when mixing height and slant.
Geometry calculators: Right Triangle & Pythagorean Calculator, Aquarium Volume Calculator, Annulus Area Calculator, Regular Polygon Calculator, Rhombus Area Calculator, Hemisphere Volume Calculator.
A pyramid holds exactly one third of the box that has the same base and height. A 6 by 4 base standing 9 tall would fill a 216 cubic unit box, and the pyramid takes 72. That one third is the same in every pyramid and cone, whatever the base.
The surface output adds the base and the four sloping faces, which is what covering it with paint, tile or shingles takes.
Measure the height straight up from the base. The slanted edge is longer, and using it in the volume formula overstates the result.
See the cone and cone frustum calculators for round versions, the rectangular prism and triangular prism pages for boxes, and the Pythagorean theorem calculator for slant heights. The rectangle area page gives the base, and the roof and stair tools in carpentry use similar geometry.
Does one third of base times height apply only to square pyramids?
No. It gives volume for a pyramid with any planar base; this tool uses a rectangular base. The displayed surface formula additionally assumes the apex is over its center.
What is the volume formula for a pyramid?
One third times the base area times the height. A 6 by 4 base and a height of 9 gives 72.
Why is a pyramid one third of a prism?
A pyramid’s volume is one third of its base area times perpendicular height. This can be established through geometric dissection in suitable cases or integration of similar cross-sections. It does not require three identical copies of every possible pyramid to fit a matching prism.
How do I find the slant height?
Use the Pythagorean theorem with the vertical height and half of the relevant base side, since the slant height is the hypotenuse of that right triangle.