Pyramid calculator
Volume, slant height and surface areas of a regular pyramid.
Fill in the fields and the result will appear here automatically.
There are two apothems in a pyramid and confusing them is the usual error. The base apothem lies flat inside the base, running from its centre to the middle of a side. The slant height is the altitude of a triangular face, measured up the sloping surface, and it is the one that enters the lateral surface area. The second is always longer than the first because it is the hypotenuse formed with the pyramid's height. The third in the volume formula is not an approximation: any pyramid or cone occupies exactly a third of the prism or cylinder standing on the same base at the same height.
How it works
Formula and logic
Base apothem = side ÷ (2 × tan(π ÷ n)). Slant height is the hypotenuse of the height and that apothem. Volume = base area × height ÷ 3.
Example
A square pyramid with a 6 cm side and 9 cm height holds 108 cm³ with a slant height of 9.487 cm.
Fields and units
- Length unit — list option
- Sides of the base — 1
- Base side length — cm
- Pyramid height — cm
How to use
- — Choose the length unit for every input.
- — Enter how many sides the base polygon has.
- — Enter the length of one base side.
- — Enter the vertical height from the base to the apex.
- — Enter an integer n from 3 to 100. Positive height is perpendicular to the base, with the apex above its centre.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: volume by integrating cross-sectional areas OpenStax: sine and tangent in a right triangle
- Limitation
- A regular base with the apex above its centre. Face slant height differs from the base apothem and a lateral edge. Total surface includes the base; thickness and allowances are excluded, and units are not automatically converted.
FAQ
Is the height measured vertically or along a face?
Vertically, from the centre of the base to the apex. The measurement along a face is the slant height, and it is returned as a result rather than taken as an input.
Why is the volume a third rather than a half?
Parallel cross-sections shrink with the square of distance from the apex. Integration gives V = Bh/3, where B is base area. Thus volume is one third of a prism with the same B and h; this does not require that three identical copies of this pyramid tile that prism.
How can I check a large square pyramid?
A hypothetical square pyramid with side 230 m and height 146 m has volume 230²·146/3 = 2,574,466.67 m³. This checks the formula for given dimensions; it is not a survey of a historical structure.
What about a pyramid whose apex is off-centre?
The volume formula still holds, but the faces are no longer identical and the single slant height stops being meaningful. This calculation assumes a regular pyramid.