Pyramid Frustum Calculator
Compute the volume, lateral surface area and total surface area of the frustum of a pyramid — the bottom part of a pyramid cut by a plane parallel to its base. Supports square and rectangular bases, with vertical height or face slant height as input.
A rectangular frustum has two different face slant heights, so vertical height h is the natural input here.
| Bottom base area A₁ | — |
| Top base area A₂ | — |
| Geometric mean √(A₁·A₂) | — |
| Face slant height m | — |
| Lateral surface area (4 trapezoidal faces) | — |
| Total surface area (A₁ + A₂ + lateral) | — |
Formulas
Let A₁ be the bottom base area, A₂ the top base area, h the vertical height, and m the face slant height (the altitude of each trapezoidal face).
- Volume: V = h/3 × (A₁ + A₂ + √(A₁·A₂))
- Square base: A₁ = s₁², A₂ = s₂²
- Rectangular base: A₁ = p₁·l₁, A₂ = p₂·l₂
- Face slant height (square): m = √(((s₁−s₂)/2)² + h²)
- Height from slant height: h = √(m² − ((s₁−s₂)/2)²)
- Lateral area (square): L = 2·(s₁+s₂)·m (four trapezoids, each (s₁+s₂)/2 × m)
- Total surface area: T = A₁ + A₂ + L
The volume formula is exactly "full pyramid minus the small top pyramid", simplified via similar triangles.
Worked examples
1. Square planter — s₁ = 4, s₂ = 2, h = 3
A₁ = 4² = 16, A₂ = 2² = 4, √(16·4) = 8.
V = 3/3 × (16 + 4 + 8) = 28.
Slant height m = √(((4−2)/2)² + 3²) = √10 ≈ 3.1623.
Lateral L = 2·(4+2)·√10 ≈ 37.9473, total T = 16 + 4 + L ≈ 57.9473.
2. Square concrete footing — s₁ = 10, s₂ = 6, h = 4
A₁ = 100, A₂ = 36, √(100·36) = 60.
V = 4/3 × (100 + 36 + 60) = 4/3 × 196 ≈ 261.3333.
m = √(2² + 4²) = √20 ≈ 4.4721.
Lateral L = 2·(10+6)·√20 ≈ 143.1084, total T ≈ 279.1084.
3. Rectangular hopper — 8 × 5 bottom, 4 × 2.5 top, h = 6
A₁ = 8·5 = 40, A₂ = 4·2.5 = 10, √(40·10) = 20.
V = 6/3 × (40 + 10 + 20) = 140.
The two face slant heights differ: m ≈ 6.1288 and ≈ 6.3246.
Lateral L ≈ 120.9801, total T ≈ 170.9801.
4. Solving from the slant height — s₁ = 9, s₂ = 3, m = 5
h = √(5² − ((9−3)/2)²) = √(25 − 9) = 4.
A₁ = 81, A₂ = 9, √(81·9) = 27.
V = 4/3 × (81 + 9 + 27) = 4/3 × 117 = 156.
Lateral L = 2·(9+3)·5 = 120, total T = 81 + 9 + 120 = 210.
5. Degenerate case: top = bottom (a box) — s₁ = s₂ = 5, h = 7
When the two bases are equal the frustum is just a prism:
V = 25 × 7 = 175, lateral L = 4·5·7 = 140, total T = 25 + 25 + 140 = 190.
Frequently asked questions
What is the frustum of a pyramid?
A frustum of a pyramid is the solid left when a pyramid is cut by a plane parallel to its base and the small top pyramid is removed. What remains has two parallel, similar bases: a larger bottom base and a smaller top base, joined by trapezoidal faces. A square pyramid frustum is described by its bottom side s1, top side s2, and vertical height h.
What is the formula for the volume of a pyramid frustum?
V = h/3 × (A1 + A2 + √(A1×A2)), where A1 and A2 are the areas of the bottom and top bases and h is the vertical height. For square bases, A1 = s1² and A2 = s2². Example: s1 = 4, s2 = 2, h = 3 gives V = 3/3 × (16 + 4 + √64) = 28.
How do you find the slant height of a pyramid frustum?
For a square frustum, the face slant height is m = √(((s1−s2)/2)² + h²) — the horizontal run is half the difference of the two sides. Example: s1 = 4, s2 = 2, h = 3 gives m = √(1 + 9) = √10 ≈ 3.1623. If you know m instead of h, invert it: h = √(m² − ((s1−s2)/2)²).
What is the lateral surface area of a pyramid frustum?
The lateral area is the sum of the trapezoidal faces, excluding the two bases. For a square frustum: L = 2×(s1+s2)×m, i.e. four trapezoids each of area (s1+s2)/2 × m. Example: s1 = 4, s2 = 2, m = √10 gives L = 12√10 ≈ 37.9473. Add both bases for the total surface area: T = A1 + A2 + L ≈ 57.9473.
What is the difference between a frustum of a pyramid and a frustum of a cone?
They are different solids with different formulas. A pyramid frustum has polygon bases — square or rectangular — and its faces are trapezoids; no π appears anywhere. A cone frustum (truncated cone) has circular bases and its formulas use the radii and π. If your solid has circular bases, use our frustum of a cone calculator instead.
How is a frustum related to a full pyramid?
A frustum is a full pyramid minus the smaller, similar pyramid that was cut off the top. The volume formula V = h/3 × (A1 + A2 + √(A1×A2)) is exactly that subtraction, simplified with similar triangles. As a check, a full pyramid (top shrunk to a point, A2 = 0) gives V = h/3 × A1, the familiar pyramid volume.
What happens when the top base equals the bottom base?
The frustum degenerates into a prism (a box): with s1 = s2 = s, the formula gives V = h/3 × (s² + s² + s²) = s²h, the ordinary box volume. Example: s1 = s2 = 5, h = 7 gives V = 175, lateral area 140, total surface area 190.
How do you compute a rectangular (non-square) pyramid frustum?
Use A1 = p1×l1 and A2 = p2×l2 in the same volume formula V = h/3 × (A1 + A2 + √(A1×A2)). The lateral area is two pairs of trapezoids with different face slant heights. Example: an 8×5 bottom, 4×2.5 top, h = 6 gives V = 140, face slants ≈ 6.1288 and ≈ 6.3246, lateral area ≈ 120.9801, total ≈ 170.9801.
Where are pyramid frustums used in real life?
Planters and flower pots, hoppers and funnels, buckets, lampshades, concrete footings and pile caps, grain silo transitions, and 3D-printed enclosures. Any container that tapers from a larger square or rectangular opening to a smaller one is a pyramid frustum.
Can the top base be larger than the bottom base?
Yes — that is the same solid flipped upside down (an inverted frustum, like a typical flower pot). The volume and surface-area formulas are symmetric in the two bases, so just enter the two side lengths; the result is identical either way.
See also
- Frustum of a Cone Calculator — truncated cone: circular bases, radius-based formulas with π.
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