Volume of a Pyramid Calculator
Calculate the volume, surface area, and slant height of a rectangular base pyramid from its base dimensions and height. See also Volume of Cone Calculator and Volume of Triangular Prism Calculator.
How to Calculate the Volume of a Pyramid
To find the volume of a rectangular base pyramid, multiply the base area (length × width) by the perpendicular height, then divide by 3. The pyramid volume is exactly one-third of a rectangular prism with the same base and height. For a square base pyramid, the length and width are equal. This calculator also computes the total surface area and slant heights.
Pyramid Volume Formula
V = (1/3) × l × w × h
Base Area = l × w
Slant Height (along l) = √(h² + (w/2)²)
Slant Height (along w) = √(h² + (l/2)²)
SA = l×w + l×sl + w×sw
(where sl and sw are the respective slant heights)
Example
Find the volume of a pyramid with base 6×6 and height 8:
V = (1/3) × l × w × h
V = (1/3) × 6 × 6 × 8
V = (1/3) × 288
V = 96 cubic units
Pyramid Volume Reference Table
| Base (L × W) | Height | Volume | Surface Area |
|---|---|---|---|
| 3 × 3 | 4 | 12.0000 | 34.6320 |
| 4 × 4 | 5 | 26.6667 | 59.0813 |
| 5 × 5 | 6 | 50.0000 | 90.0000 |
| 6 × 6 | 8 | 96.0000 | 138.5280 |
| 6 × 4 | 8 | 64.0000 | 107.6533 |
| 8 × 6 | 10 | 160.0000 | 196.1444 |
| 8 × 8 | 10 | 213.3333 | 236.3253 |
| 10 × 8 | 12 | 320.0000 | 310.4911 |
| 10 × 10 | 12 | 400.0000 | 360.0000 |
| 10 × 10 | 15 | 500.0000 | 416.2278 |
| 12 × 10 | 15 | 600.0000 | 471.2916 |
| 12 × 12 | 16 | 768.0000 | 554.1122 |
| 15 × 10 | 12 | 600.0000 | 486.5097 |
| 15 × 15 | 20 | 1500.0000 | 865.8003 |
| 20 × 20 | 25 | 3333.3333 | 1477.0330 |
Frequently Asked Questions
What is the volume of a pyramid?
The volume of a pyramid is the total three-dimensional space enclosed within its base and triangular faces meeting at the apex. For a rectangular base, V = (1/3) × length × width × height.
Why is the pyramid volume one-third of a prism?
This can be proven using calculus or by decomposing a rectangular prism into three pyramids of equal volume. The factor of 1/3 applies to all pyramids regardless of base shape.
What is the slant height?
The slant height is the distance from the apex to the midpoint of a base edge, measured along the triangular face. A rectangular pyramid has two different slant heights — one for each pair of opposite faces.
Does this work for a square pyramid?
Yes. A square pyramid is a special case where the base length and width are equal. Enter the same value for both to calculate a square pyramid.
What about triangular or hexagonal base pyramids?
The general formula V = (1/3) × base area × height works for any base shape. This calculator specifically handles rectangular bases. For other bases, calculate the base area separately and multiply by h/3.