What is a pyramid?
A pyramid is a solid with a polygonal base and a single apex connected to the base by triangular faces. The perpendicular height h is measured straight from the apex to the base plane; it is not the same thing as the slant height of a face.
Volume
Take the base area B, multiply by the perpendicular height h, then divide by 3.
Surface area
Total surface combines the base area with the lateral area of the triangular faces.
Why the factor 1/3 matters
For any pyramid, the volume is one-third of a prism with the same base area and perpendicular height. That is why V = (1/3)Bh works even when the base is not a square. This general relationship is also the reason pyramid volume parallels the one-third factor used for a cone.
Pyramid formulas, step by step
The volume formula needs only base area and perpendicular height. Surface-area formulas need the geometry of the sloping faces, so the slant height must be derived first.
- V
- pyramid volume
- B
- area of the base
- h
- perpendicular height
Rectangular base
For base length a and width b.
Square base
A square is the special case where a = b = s.
Slant height of a rectangular pyramid
For a right rectangular pyramid, there are generally two face slant heights. Across the faces whose base edge is a, the triangular-face height is based on half of b; across faces whose base edge is b, it is based on half of a.
- L
- lateral surface area of the four triangular faces
- a,b
- base side lengths
- ℓₐ,ℓᵦ
- corresponding face slant heights
Truncated pyramid (frustum)
If the apex is cut parallel to the base, the result is a frustum. With bottom dimensions a × b, top dimensions a′ × b′ and height h, the volume uses the geometric-mean relationship between the two rectangular base areas.
Pyramid examples with full reasoning
Example 1 — Square pyramid volume
Suppose a square pyramid has side length s = 6 m and perpendicular height h = 4 m. Its base area is 6² = 36 m².
Example 2 — Slant height and surface area
For the same square pyramid, the face slant height is the hypotenuse of a right triangle with legs h = 4 and s/2 = 3.
Example 3 — Rectangular pyramid
Take a = 8 m, b = 5 m and h = 9 m. The base area is 40 m², so the volume is:
Example 4 — Truncated square pyramid
Let the bottom side be 6 m, the top side 2 m and the perpendicular height 4 m. The square-frustum formula gives:
Where pyramid calculations are useful
Architecture
Pyramidal roofs, monuments, skylights and tapered structural elements use the same base-area, height and slant relationships.
Packaging
Pyramid-shaped packages and displays use volume for capacity and lateral area for material estimates.
Construction
Surface area helps estimate cladding, paint, membrane or sheet material on sloping faces.
Education
The pyramid is an excellent model for connecting area, volume, Pythagorean theorem and three-dimensional reasoning.
Pyramid calculator questions
Find the base area B, multiply it by the perpendicular height h, then divide by 3: V = ⅓Bh.
The perpendicular height goes from the apex to the base plane at 90°. The slant height runs along a triangular face. They are different lengths except in special limiting situations.
For a right rectangular pyramid, calculate the two corresponding face slant heights, then use L = aℓₐ + bℓᵦ and add the base area ab for total surface area.
Yes. The general pyramid volume relationship still uses one-third of base area times perpendicular height. The slant/surface geometry becomes different when the apex is not above the base center.
You get a square pyramid frustum. Its volume depends on the bottom area, top area and the geometric-mean term between them.
If lengths are measured in meters, area is in m² and volume is in m³. The calculator preserves the mathematical units implied by your input dimensions.