Pyramid Frustum Calculator
Calculate volume and surface area of a pyramid frustum (truncated square pyramid) from top side, bottom side, and height.
Includes slant height formula.
Pyramid Frustum (Truncated Square Pyramid)
A pyramid frustum is formed by cutting a pyramid with a plane parallel to the base. This calculator handles the square pyramid frustum — where both the top and bottom faces are squares.
Formulas:
| Property | Formula |
|---|---|
| Volume | V = (h/3)(a^2 + ab + b^2) |
| Slant Height | l = sqrt(h^2 + ((b-a)/2)^2) |
| Lateral Surface Area | LSA = 2(a + b) * l |
| Top Area | A_top = a^2 |
| Bottom Area | A_bot = b^2 |
| Total Surface Area | TSA = a^2 + b^2 + 2(a+b)*l |
Variables:
- a = top side length (smaller square)
- b = bottom side length (larger square)
- h = perpendicular height between the two bases
- l = slant height of each trapezoidal face
Worked example — stepped pedestal (a = 2 m, b = 5 m, h = 3 m):
- Slant height: l = sqrt(9 + 2.25) = sqrt(11.25) ≈ 3.354 m
- Volume: (3/3)(4 + 10 + 25) = 39 m^3
- Lateral SA: 2(2+5)(3.354) = 2(7)(3.354) ≈ 46.96 m^2
- Total SA: 4 + 25 + 46.96 ≈ 75.96 m^2
Special cases:
- If a = 0: frustum becomes a complete pyramid — Volume = hb^2/3 ✓
- If a = b: frustum becomes a square prism — Volume = ha^2 ✓
- The formula is derivable from the prismatoid formula: V = (h/6)(A_top + 4A_mid + A_bot) where A_mid = ((a+b)/2)^2
Real-world pyramid frustums: The base of the Washington Monument is a frustum shape. Ancient Mesoamerican step pyramids, the bases of obelisks, tapered building columns, and some crystal structures all follow this geometry. The frustum also appears in architecture when a tower narrows as it rises.