Pyramid Volume Calculator
Volume of a square pyramid from base side and height, in metric and imperial.
What it does
A square pyramid rises from a square base to a single apex. Its volume is one third of the box that shares the same base and height, since three such pyramids fill that prism exactly.
The tool above takes the base side with the vertical height in centimeters or inches. It returns the volume and the base area. The default side 6 with height 9 gives base 36 cm2 and volume 108 cm3.
Calculus and prealgebra texts state the same rule as V = 1/3 a squared h for square bases, with cones as the round twin at V = 1/3 pi r squared h. The one third factor is the whole story for every pointed solid over its matching straight solid.
Height means the perpendicular drop from apex to base center. Slant edges and face altitudes run longer, and feeding them into the formula overstates the answer; measure straight down instead.
How to use
- Pick centimeters or inches.
- Type the square base side.
- Type the vertical height to the apex.
- Read volume first, base area beside it.
- Check: volume times 3 divided by height must equal base area.
- Convert mirrors land on the same volume through 16.387064.
Formula
- a
- Square base side length.
- h
- Vertical height from base to apex.
- B
- Base area, side squared.
Imperial inputs convert to centimeters first (x 2.54), then divide cubic centimeters by 16.387064 for cubic inches, so both systems agree.
Worked example
Example: side 6, height 9
Metric defaults: side 6 cm, height 9 cm.
- Base: 6 x 6 = 36 cm2.
- Multiply: 36 x 9 = 324.
- Third: 324 / 3 = 108 cm3.
- Prism check: the matching box holds 324, one third is 108.
- Imperial check: 2.362 by 3.543 in gives 6.59 in3.
- Tool shows 108 cm3 with base 36 cm2.
Reference table
| Side and height | Base | Volume |
|---|---|---|
| 6 and 9 cm | 36 cm2 | 108 cm3 |
| 4 and 9 cm | 16 cm2 | 48 cm3 |
| 3 and 6 cm | 9 cm2 | 18 cm3 |
| Cone r 2, h 6 in | 12.57 in2 | 25.12 in3 |
Limits
- Square bases only; other bases need their own area.
- Right pyramids assumed; oblique apexes shift little but differ.
- Vertical height required, never slant edge.
- Hollow walls need shell math, not solid volume.
Common mistakes
- Skipping the divide by 3.
- Feeding slant height as vertical height.
- Squaring the height instead of the side.
- Mixing cm and inches in one calculation.
- Reporting square units for a volume.
Try it yourself
Quick checks. Answers sit tucked under each question.
Side 4, height 9?
Base 16 cm2, volume 48 cm3.
Side 3, height 6?
Base 9 cm2, volume 18 cm3.
Pyramid versus prism?
One third: the prism holds 3 times the pyramid.
Questions
What is the pyramid formula?
Base area times height over 3; square base gives a squared times h over 3.
Why one third?
Three pyramids fill their matching prism, by slicing or rearrangement.
Slant or vertical height?
Vertical only. Slant edges run longer and overstate volume.
Cone twin?
Same third rule: 1/3 pi r squared h. A r 2 h 6 cone holds 25.12 cubic inches.
Do units convert?
Yes. 108 cm3 is 6.59 in3 through 16.387064.
What if the base is rectangular?
Base area is length times width then; the divide by 3 stays.