The Truncated Pyramid Calculator computes the volume and lateral surface area of a frustum using your top and bottom dimensions and height. It also handles common units so your results stay consistent.
This guide explains the exact formulas, what each dimension means, and how to avoid the two most common mistakes: mixing units and confusing slant height with vertical height.
What Is a Truncated Pyramid (Frustum)?
A truncated pyramid (also called a frustum) is a pyramid with the top cut off parallel to the base. The result has two similar, parallel faces (top and bottom) and four trapezoidal side faces.
When the pyramid sides are straight and the cut is parallel, the frustum keeps the shape’s key property: linear dimensions scale consistently between the top and bottom faces.
Core Inputs You Need
To compute volume and surface area, you need:
- Height (h): the vertical distance between the top and bottom faces.
- Top base dimensions: width a and length b (rectangle) or a single side length if square.
- Bottom base dimensions: width A and length B (rectangle) or a single side length if square.
For surface area, you also need the slant height along the direction of each side. For a rectangular frustum, the calculator uses a practical method based on the horizontal offsets and height.
Key Formulas (Simple and Exact)
1) Volume of a Truncated Pyramid
Let the areas of the top and bottom rectangles be:
- Top area At = a × b
- Bottom area Ab = A × B
The volume is:
V = h / 3 × (Ab + At + √(Ab × At))
This formula comes from integrating the similarity scaling between cross-sections.
2) Lateral Surface Area of a Rectangular Frustum
The lateral surface area is the sum of the four trapezoid side areas. For each direction, we compute a slant height and multiply by the corresponding average side length.
Define half-differences in each direction:
- Horizontal offset in width direction: dx = (A − a) / 2
- Horizontal offset in length direction: dy = (B − b) / 2
Then the slant heights are:
- sx = √(h² + dx²)
- sy = √(h² + dy²)
The lateral area becomes:
L = (a + A) × b_avg × sx + (b + B) × a_avg × sy
But for rectangular frustums, the trapezoid area decomposition is clearer:
- Two sides parallel to length: each has area (a + A) / 2 × b × sx when paired with the matching dimension.
- Two sides parallel to width: each has area (b + B) / 2 × a × sy when paired with the matching dimension.
The calculator uses the consistent trapezoid approach so you get reliable lateral area without guessing.
3) Total Surface Area (Optional)
If you also want the full surface area, add the top and bottom areas:
SA = Ab + At + L
This is useful for packaging, insulation, and material estimates.
How Unit Conversion Works
Volume depends on length³, while surface area depends on length². That means:
- Changing length units changes volume by a cube factor.
- Changing length units changes area by a square factor.
The calculator converts all inputs to a consistent internal unit, then converts the outputs to your selected output unit.
Practical Examples (Real Use-Cases)
Example 1: Designing a Tapered Container
Suppose you’re building a tapered container where the bottom is 40 cm × 30 cm, the top is 25 cm × 18 cm, and the height is 20 cm. You need the volume to choose the right material thickness and capacity.
Enter those dimensions and height in the same unit. The calculator returns volume immediately, so you can compare alternatives (different cut sizes) without re-deriving formulas.
Example 2: Estimating Material for a Chute or Funnel
If you’re covering a truncated pyramid chute, you care about lateral surface area to estimate how much sheet material you need. For a bottom of 60 in × 40 in, top of 30 in × 20 in, and height of 24 in, the calculator gives lateral area for the side panels.
Use total surface area if you also plan to include the top and bottom faces.
Common Mistakes to Avoid
- Mixing units: always input all dimensions using the same unit selection.
- Using slant height directly: you typically measure vertical height (h). Slant height is computed from h and the horizontal offsets.
- Assuming the frustum is square: if it’s rectangular, use both width and length.
Frequently Asked Questions
What is the formula for the volume of a truncated pyramid?
For a rectangular frustum, compute the top area At = a×b and bottom area Ab = A×B. Then use V = h/3 × (Ab + At + √(Ab×At)). This works when the top cut is parallel to the base, keeping similarity.
How do I find the slant height for a truncated pyramid?
Slant height depends on the horizontal offset between top and bottom in the direction of the side. For width, dx = (A−a)/2 and sx = √(h² + dx²). For length, dy = (B−b)/2 and sy = √(h² + dy²).
Is lateral surface area the same as total surface area?
No. Lateral surface area counts only the four side trapezoids. Total surface area adds the top and bottom faces: SA = Ab + At + L. Use lateral area for side covering and total area for full material wrap.
Can I use this calculator for a square truncated pyramid?
Yes. A square frustum is a special case where width equals length at both top and bottom. Enter the same value for a and b (top) and for A and B (bottom). The formulas still apply because the areas are computed consistently.
What units should I use for volume and surface area outputs?
Volume is reported in cubic units (like cm³ or in³), and surface area is reported in square units (like cm² or ft²). The calculator converts your chosen input unit into consistent internal units, then converts results to your selected output unit.
Next Steps: Verify Your Measurements
Before using the results, confirm that your top and bottom faces are parallel and that your height is the vertical distance between them. If the cut is angled or not parallel, the standard frustum formulas will not match reality.
With correct inputs, the Truncated Pyramid Calculator gives fast, accurate volume and surface area for engineering, DIY, and construction planning.
