Pyramid Volume
Calculator
Results
- Volume
- 108
- Slant height
- 9.486832
- Lateral surface area
- 113.841995
- Base area
- 36
- Total surface area
- 149.841995
Results
| Volume | 108 |
| Slant height | 9.486832 |
| Lateral surface area | 113.841995 |
| Base area | 36 |
| Total surface area | 149.841995 |
formula-map diagram
- Volume
- 108
- Slant height
- 9.486832
- Lateral surface area
- 113.841995
- Base area
- 36
- Total surface area
- 149.841995
Formula map
Formula
V = (1/3) × a² × h, A = a² + 2 × a × √(h² + (a/2)²)= 108
Note
Simplified model: these are the exact results of the stated idealised geometric formulas, assuming perfect shapes, exact inputs and consistent units. Real measurements carry error, and results such as the cyclic-quadrilateral area only hold when the figure truly satisfies the stated condition. Verify independently before relying on these figures.
More in Advanced geometry
See all →Frequently asked questions
What is the formula for the volume of a pyramid?+
Volume equals one-third times the base area times the height: V = (1/3)Bh, where B is the area of the base (whatever its shape) and h is the perpendicular height from the base to the apex. The one-third factor holds for any pyramid or cone, regardless of the base shape.
Why is a pyramid's volume exactly one-third of the equivalent prism?+
A prism with the same base and height has volume V = Bh, and it can be shown geometrically (and via calculus integration) that three congruent pyramids of that base and height exactly fill that prism. This one-third relationship is a fundamental geometric fact, not an approximation.
What height should I use if the pyramid is not a right pyramid (its apex isn't centered)?+
Always use the perpendicular height, the shortest straight-line distance from the apex directly down to the plane of the base, not the slant height along an edge or face. This holds true even for oblique pyramids where the apex sits off to one side.
How is surface area different from volume for a pyramid, and why does the calculator give both?+
Volume measures the space enclosed inside the pyramid, while surface area sums the base area plus the areas of all the triangular lateral faces, using the slant height rather than the vertical height. Both are useful but for very different purposes, volume for capacity or material fill, surface area for material needed to cover the outside.
Does the base shape (square, triangular, hexagonal) change the volume formula?+
No, the (1/3)Bh formula works for any base shape whatsoever, since B simply represents that shape's own area, calculated with whatever formula fits a square, triangle, hexagon, or irregular polygon. Only the method of computing B changes; the one-third relationship to height stays constant.