Truncated Cone Calculator — Frustum Surface Area & Volume
A truncated cone (frustum) is a cone with the pointed tip cut off by a plane parallel to the base, leaving two circular faces. Enter the bottom radius, top radius and height to calculate volume, lateral surface area, total surface area and slant height.
units
units
units
Lateral surface area + both circular bases
410.5 u³
Volume- 1
Slant height s = √(h² + (r₁ − r₂)²)
√(8² + (5 − 3)²) = 8.2462 - 2
Lateral surface area π(r₁ + r₂) × s
π × (5 + 3) × 8.2462 = 207.2499 - 3
Bottom base area π × r₁²
π × 5² = 78.5398 - 4
Top base area π × r₂²
π × 3² = 28.2743 - 5
Total surface area
207.2499 + 78.5398 + 28.2743 = 314.0640Lateral surface plus both circular caps.
How does this calculator work?
Frustum: V = (π/3)h(r₁²+r₁r₂+r₂²); lateral area = π(r₁+r₂)s where s = √(h²+(r₁−r₂)²); total area = lateral area + πr₁² + πr₂². Reduces to a cone when r₂=0, to a cylinder when r₁=r₂.
Formula
How this is calculated
A frustum is the portion of a solid (usually a cone) that lies between two parallel planes cutting it. The two circular faces have radii r₁ (bottom, larger) and r₂ (top, smaller), and the perpendicular distance between them is the height h. The slant height s is the length of the slanting edge: s = √(h² + (r₁ − r₂)²), derived from the Pythagorean theorem applied to the cross-section.
The volume formula V = (π/3) × h × (r₁² + r₁r₂ + r₂²) can be derived by computing the difference between two complete cones — the original full cone and the small cone removed from the top — and simplifying. It reduces correctly to a full cone (V = πr²h/3) when r₂ = 0, and to a cylinder (V = πr²h) when r₁ = r₂.
The lateral (curved) surface area is A_lat = π(r₁ + r₂) × s, the area of the band wrapping the slant surface. Adding the two circular base areas (πr₁² + πr₂²) gives the total surface area. Common applications include buckets, drinking cups, lampshades, silos, and many engineering components.
Frequently asked questions
They are the same shape — "frustum" is the general mathematical term for any solid between two parallel planes cutting a pyramid or cone. "Truncated cone" specifically means the frustum derived from a right circular cone. Both names are correct and interchangeable in everyday use.
When r₁ = r₂, the shape becomes a cylinder. The slant height equals the height (s = h, since |r₁ − r₂| = 0), the lateral area formula gives 2πrh (the cylinder's lateral area), and the volume formula gives πr²h — both correct. This calculator handles that case correctly.
Rearrange the volume formula: h = 3V / [π(r₁² + r₁r₂ + r₂²)]. If both radii are known, divide three times the volume by π times the bracket quantity. There is no closed-form solution if the radii are also unknown — you would need at least two additional constraints.
Also known as
TG we-Calculate Editorial Team. (2026). Truncated Cone Calculator — Frustum Surface Area & Volume [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/truncated-cone-calculator
TG we-Calculate Editorial Team. "Truncated Cone Calculator — Frustum Surface Area & Volume." TG we-Calculate. 2026. https://we-calculate.com/calculator/truncated-cone-calculator.
TG we-Calculate Editorial Team, "Truncated Cone Calculator — Frustum Surface Area & Volume," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/truncated-cone-calculator
@misc{wecalculate_truncated_cone_calculator, title = {Truncated Cone Calculator — Frustum Surface Area & Volume}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/truncated-cone-calculator}}, year = {2026}, note = {TG we-Calculate} }
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