Prisms and Pyramids
A prism is a solid with two parallel, congruent polygon bases joined by parallelogram faces, and a pyramid is a solid with one polygon base and triangular faces meeting at a point called the apex. In brief: the volume of a prism is V = B · h (base area times height) and the volume of a pyramid is V = (1/3) B · h. A square pyramid with base side 6 and height 4 has volume (1/3) · 36 · 4 = 48.
Prisms and pyramids are the basic polyhedra of school geometry. Their volumes and surface areas follow from the area of the base and a few lengths, and they appear in buildings, packaging and crystal shapes. This guide explains the parts and types, the formulas for volume and surface area (including the slant height of a regular pyramid), worked examples and a table that lists the solids by the shape of the base.
What is a prism?
A prism has two parallel bases that are congruent polygons. The other faces, the lateral faces, are parallelograms (rectangles in a right prism). The prism is named by its base: triangular, rectangular, pentagonal, and so on. In a right prism the lateral edges are perpendicular to the bases, and the height equals the length of a lateral edge. In an oblique prism the edges are slanted, and the height is the perpendicular distance between the bases. A cube is a right prism with square bases and equal edges, and a cuboid (rectangular box) is a prism with rectangular bases.
What is a pyramid?
A pyramid has a polygonal base and triangular lateral faces that meet at the apex. The height h is the perpendicular distance from the apex to the plane of the base. A regular pyramid has a regular polygon as the base and its apex lies directly above the center of the base. Its lateral faces are congruent isosceles triangles, and the height of one of these triangles, from the apex to the midpoint of a base edge, is the slant height s. A tetrahedron is a pyramid with a triangular base, and it has four triangular faces.
What are the formulas?
Prism with base area B, base perimeter P, height h (right prism):
- Volume V = B · h
- Lateral surface area L = P · h
- Total surface area S = 2B + P · h
Regular pyramid with base area B, base perimeter P, height h and slant height s:
- Volume V = (1/3) B · h
- Lateral surface area L = (1/2) P · s
- Total surface area S = B + (1/2) P · s
- Relationship: s2 = h2 + r2, where r is the apothem of the base (distance from the center to the midpoint of a side). For a square base with side a: s2 = h2 + (a/2)2.
The volume factor 1/3 can be explained by cutting a cube into three congruent pyramids that meet at one vertex: each has the volume (1/3) of the cube, which equals (1/3) · base · height. Cavalieri’s principle extends the result to every pyramid.
Worked examples
Example 1: triangular prism
The base is a right triangle with legs 3 and 4 (hypotenuse 5), and the prism has a height of 10. Base area B = 6, perimeter P = 12. Volume V = 60. Lateral area L = 12 · 10 = 120. Total surface area S = 12 + 120 = 132.
Example 2: hexagonal prism
Regular hexagon with side 4, height 9. B = (3√3/2) · 16 = 24√3 ≈ 41.57. V = 41.57 · 9 ≈ 374.1. P = 24, L = 216, S = 83.14 + 216 = 299.1.
Example 3: square pyramid
Base side 6, height 4. B = 36, V = 48. The apothem of the base is 3, so the slant height is s = √(16 + 9) = 5. The lateral area is (1/2) · 24 · 5 = 60. Total surface area S = 36 + 60 = 96.
Example 4: edge of a pyramid
The lateral edge of this pyramid goes from the apex to a base vertex. The half-diagonal of the base is 3√2, so the edge is √(16 + 18) = √34 ≈ 5.83.
Example 5: finding the height from the volume
A pyramid with a rectangular base 5 by 9 has a volume of 120. Then (1/3) · 45 · h = 120, so h = 8.
Example 6: the regular tetrahedron
With edge a, all four faces are equilateral triangles. The surface area is 4 · (√3/4)a2 = √3 a2 and the volume is a3√2/12. For a = 6: S = 36√3 ≈ 62.35, V = 216 · 1.4142/12 ≈ 25.46. The height is a√(2/3) ≈ 4.90.
Counting vertices, edges and faces
| Solid | Vertices V | Edges E | Faces F | V − E + F |
|---|---|---|---|---|
| n-gonal prism | 2n | 3n | n + 2 | 2 |
| n-gonal pyramid | n + 1 | 2n | n + 1 | 2 |
| Triangular prism (n = 3) | 6 | 9 | 5 | 2 |
| Cube (n = 4 prism) | 8 | 12 | 6 | 2 |
| Hexagonal prism (n = 6) | 12 | 18 | 8 | 2 |
| Tetrahedron (n = 3 pyramid) | 4 | 6 | 4 | 2 |
| Square pyramid (n = 4) | 5 | 8 | 5 | 2 |
The last column illustrates Euler’s formula V − E + F = 2, valid for all convex polyhedra.
Where do these solids come from?
The volume of a pyramid is mentioned in Egyptian and Babylonian problems: the Moscow Papyrus has a problem on the volume of a truncated pyramid with a correct formula. Euclid’s Elements (Book XII) proves that a pyramid has one third of the volume of a prism with the same base and height, using the method of exhaustion, a result attributed to Eudoxus. The Egyptian pyramids are among the best-known pyramids in history.
Where are they used today?
Prisms appear in buildings, storage containers, optical prisms that split light and in crystal shapes. Pyramids are used in roofs, monuments and tents. Packaging designers compute surface areas to minimise material, and engineers compute volumes for concrete: a pyramid-shaped hopper with base 2 m by 2 m and height 3 m holds (1/3) · 4 · 3 = 4 m3.
Which mistakes are common?
Forgetting the factor 1/3 for pyramids
A pyramid has one third of the volume of the prism with the same base and height.
Using the slant height as the height in the volume formula
The volume uses the vertical height h, and the surface area uses the slant height s.
Forgetting the bases in the total surface area
The lateral area P · h excludes the two bases (for a prism), and the pyramid’s lateral area excludes its base.
Using the oblique edge as the prism height
For an oblique prism, the height is the perpendicular distance between the bases.