Pyramid frustum calculator
Volume, slant height and surfaces of a square pyramid frustum.
Fill in the fields and the result will appear here automatically.
This pyramid frustum has parallel square bases with a common central axis. Both base side lengths and perpendicular height are entered in centimetres. Cross-section side length changes linearly and area quadratically, giving V = h(a²+ab+b²)/3. Averaging the two base areas and multiplying by height overstates volume by h(a−b)²/6. Slant height follows the middle of a lateral face and differs from vertical height.
How it works
Formula and logic
Volume h/3·(S₁ + S₂ + √(S₁·S₂)); slant height √(h² + ((a−b)/2)²); lateral area 2·(a+b)·slant height.
Example
For a = 10 cm, b = 6 cm and h = 8 cm, volume is 522.67 cm³, slant height 8.246 cm, lateral area 263.88 cm² and total area 399.88 cm². Base areas are 100 and 36 cm².
Fields and units
- Bottom base side — cm
- Top base side — cm
- Height — cm
How to use
- — Enter positive side lengths of the two square bases and height in centimetres.
- — Bases must be parallel, similar and centred on one axis; height is perpendicular to them.
- — Either base order is accepted, including a larger upper base. Equal sides are rejected here as the prism case.
- — Volume is in cm³; surfaces and the base-area pair are in cm². Divide cm³ by 1000 for litres.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: volume by integrating cross-sectional areas OpenStax: sine and tangent in a right triangle
- Limitation
- Square centred bases, positive unequal sides and perpendicular height. Input unit is fixed: centimetres. The geometric formula excludes wall thickness, material allowance, reinforcement and construction tolerances.
FAQ
Why not average the base areas and multiply by the height?
With a = 10, b = 6 and h = 8 cm, averaging areas gives 544 cm³ while geometric volume is 522.67 cm³. Their difference h(a−b)²/6 = 21.33 cm³ is positive. Intermediate cross-sections are smaller than linear interpolation of the base areas.
How does slant height differ from height?
The height runs along the axis, the slant height along the middle of a face from edge to edge. The slant height is always longer, and it is what the face area is computed from.
What about rectangular bases?
For a pyramid frustum with similar parallel rectangular bases, volume is also h(S₁+√(S₁S₂)+S₂)/3. Arbitrary rectangles without one shared scale factor do not meet that condition; surface area needs two different slant heights.
Where does this shape appear?
Foundation pads, hoppers and funnels, lampshades, and classical architecture from ziggurats to pedestals. The volume is for concrete, the lateral area for cladding.