Frustum Volume Calculator – Truncated Cone and Pyramid
A frustum is the three-dimensional solid that remains after a cone or pyramid has its tip removed by a cut parallel to the base. The word comes from the Latin frustum, meaning "morsel" or "bit cut off." Frustums are everywhere in everyday life: buckets, drinking cups, grain silos, lamp shades, traffic cones, and hopper funnels are all frustum shapes. This calculator handles both the circular frustum (truncated cone) and the square frustum (truncated square pyramid), and it also computes the slant height and surface areas alongside the volume.
Circular Frustum (Truncated Cone)
A circular frustum has two circular faces of different radii. If the bottom radius is R, the top radius is r, and the perpendicular height between the bases is h, the volume formula is:
V = (π × h / 3) × (R² + R×r + r²)
This formula is a generalisation of the cone volume formula V = (1/3)πR²h. When r = 0, the expression reduces to the cone formula exactly. When r = R, it becomes the cylinder volume formula πR²h, confirming that a cylinder is the limiting case of a frustum with equal radii.
Square Frustum (Truncated Square Pyramid)
A square frustum has two square faces. If the bottom side length is a, the top side length is b, and the perpendicular height is h, the volume is:
V = (h / 3) × (a² + a×b + b²)
This is Archimedes' prismatoid formula applied to a quadrilateral frustum. When b = 0, the formula gives the pyramid volume (h/3) × a². When b = a, it becomes the rectangular prism volume a² × h.
Slant Height
The slant height is the distance measured along the outer surface from the rim of the top base to the rim of the bottom base. For a circular frustum:
l = √(h² + (R − r)²)
For a square frustum:
l = √(h² + ((a − b) / 2)²)
The slant height is essential for material estimation — for example, calculating how much sheet metal is needed to fabricate a hopper or how much fabric covers a lamp shade.
Surface Areas
The calculator provides two surface area measurements. The lateral surface area is the area of the sloped side only. For a circular frustum:
LSA = π × (R + r) × l
For a square frustum, the four trapezoidal faces have a combined area of:
LSA = 2 × (a + b) × l
The total surface area adds both base areas:
TSA = LSA + area of bottom base + area of top base
Real-World Applications
Engineers use frustum volume calculations when designing storage tanks, hoppers, and silos to determine capacity. Architects apply the formulas to roof sections, decorative elements, and structural members. In 3D printing, slicing software computes frustum geometry to estimate material usage for tapered objects. Civil engineers rely on frustum calculations for earthworks — estimating cut-and-fill volumes for roads and foundations.
The formulas in this calculator are exact for perfect geometric frustums. For real-world constructions with wall thickness, material offsets, or non-parallel cuts, you would apply these results as a baseline and adjust for the specific geometry.
Unit Conversions
The tool accepts five length units — millimetres, centimetres, metres, inches, and feet — and displays the volume result in your chosen unit cubed. It simultaneously converts the volume to cubic metres (m³), cubic centimetres (cm³), litres (L), and cubic feet (ft³). A practical alternate unit is also shown: mL for centimetre inputs, US gallons for foot inputs, and US fluid ounces for inch inputs, making it easier to relate the result to everyday measurements.
Tips for Getting Accurate Results
Use consistent units for all three input dimensions — mixing centimetres and inches will give incorrect results. When measuring a physical object, measure the perpendicular height, not the slant height; the calculator computes the slant height from the perpendicular height you supply. For the top dimension, if the original cone or pyramid apex is fully intact, enter 0 to compute the full cone or pyramid volume — the tool handles this case automatically.