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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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