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Truncated Cone Calculator

A truncated cone (or frustum) is the portion of a cone that remains after slicing off the apex with a plane parallel to the base. Engineers, architects, and students frequently need to find volumes, surface areas, and slant heights for these shapes—whether designing tapered funnels, lamp bases, or ventilation ducts. This calculator solves any truncated cone problem by accepting whatever measurements you know and computing the rest.

Last updated:

Creators Jasmine J. Mah, BSc
3,580 people find this calculator helpful

Understanding the Truncated Cone

A truncated cone forms when you cut a complete cone horizontally, removing the pointed tip while keeping the circular top and base parallel. This geometry appears constantly in real applications: paper cups, bucket shapes, traffic cones, and industrial hoppers all rely on frustum proportions.

The key dimensions are:

  • Height (h): perpendicular distance between the two circular faces
  • Base radius (R): radius of the larger circular base
  • Top radius (r): radius of the smaller circular top
  • Slant height (s): distance measured along the lateral surface from rim to rim

Understanding these parameters is essential before calculating any derived properties like volume or surface area.

Core Formulas for Truncated Cones

The relationships between dimensions and properties follow from geometric principles. Slant height connects to the radii and height via the Pythagorean theorem, while volume integrates the cross-sectional areas, and surface area sums the three regions: lateral face, top circle, and base circle.

Volume: V = (1/3) × π × h × (r² + r×R + R²)

Slant height: s = √((R − r)² + h²)

Lateral surface area: A_lateral = π × (r + R) × s

Top area: A_top = π × r²

Bottom area: A_bottom = π × R²

Total surface area: A_total = π × (r + R) × s + π × r² + π × R²

  • h — Height of the truncated cone (perpendicular distance between circular faces)
  • r — Radius of the top (smaller) circular face
  • R — Radius of the base (larger) circular face
  • s — Slant height measured along the lateral surface
  • V — Volume enclosed by the truncated cone
  • A_lateral — Area of the curved side surface
  • A_top — Area of the top circular face
  • A_bottom — Area of the base circular face
  • A_total — Sum of all external surface areas

How to Use This Calculator

Enter any combination of known values—you might have height and both radii, or perhaps volume and slant height. The calculator identifies which measurement you've provided and instantly computes all unknowns. If contradictory inputs trigger a warning, double-check your entries: they may violate geometric constraints (for example, requesting a height shorter than the radius difference).

Unit flexibility is built in, so switch between millimetres, centimetres, metres, inches, or feet as needed. The output respects your chosen units throughout: cubic units for volume, square units for areas.

Common Pitfalls and Practical Considerations

Avoid these mistakes when working with truncated cone calculations.

  1. Confusing slant height with vertical height — Slant height (distance along the curved surface) is always longer than the perpendicular height unless the radii are equal. Never substitute one for the other. Use the Pythagorean theorem to derive slant height from height and radius difference if needed.
  2. Forgetting to include top and bottom in total surface area — Total surface area comprises three parts: the lateral (curved) surface plus two circular faces. If you only need the lateral area for a single calculation, that's fine—but for total area, account for all three components.
  3. Unit mismatches in formulas — If your measurements mix centimetres and metres, convert everything to the same unit before calculating. Volume units will be cubic (cm³ or m³), and area units will be square (cm² or m²). Inconsistent units lead to nonsensical results.
  4. Negative radius differences yield imaginary slant heights — For a valid truncated cone, the base radius must exceed the top radius. If you enter a top radius larger than the base radius, no real solution exists. Check that R ≥ r always.

Practical Applications in Engineering and Design

Truncated cones appear across numerous industries. Manufacturing uses frustum geometry for tapered pipe fittings, reducing pressure drop in fluid systems. Acoustic design relies on frustum shapes in horn speakers to control sound dispersion. Architectural lighting fixtures—from classic chandelier shades to modern pendant lamps—often follow frustum proportions for aesthetic balance and light distribution.

In civil engineering, conical hoppers for grain storage or aggregate handling are frequently truncated to fit standard container dimensions. Calculating exact volumes ensures accurate capacity planning. Similarly, ventilation ductwork often tapers from larger trunk lines to smaller branch lines, requiring frustum volume and flow-area calculations for proper air distribution.

Frequently Asked Questions

What's the difference between a full cone and a truncated cone?

A complete cone tapers to a single point (the apex), whereas a truncated cone has that apex removed by a horizontal cut, leaving two parallel circular faces. Both shapes measure volume using integration methods, but the frustum formula accounts for both the top and base radii, making it distinct. Truncated cones appear far more often in practical applications because their flat top makes them stackable and easier to manufacture.

Can I find the slant height if I only know height and the two radii?

Yes. The Pythagorean theorem directly gives you slant height: s = √((R − r)² + h²). The height and the difference between radii form a right triangle where slant height is the hypotenuse. For example, if height is 6 and radii are 3 and 5, then s = √((5 − 3)² + 6²) = √(4 + 36) = √40 ≈ 6.32 units.

How does volume scale if I double all the dimensions of a truncated cone?

Volume increases by a factor of eight (2³). Since volume has units of length cubed, scaling all linear dimensions by a factor k multiplies volume by k³. This applies regardless of the cone's proportions. Doubling height and both radii from a 100 cm³ frustum yields 800 cm³.

What if the top and bottom radii are equal?

If r = R, you have a cylinder, not a truncated cone. The frustum formulas still work mathematically: slant height becomes just the height, lateral area simplifies to 2πRh, and volume becomes πR²h. However, a cylinder is technically a degenerate frustum with zero apex angle.

How do I calculate the lateral surface area if I have the radii but not the slant height?

First compute slant height using s = √((R − r)² + h²), then apply the lateral area formula: A = π(r + R)s. For radii 4 and 6 with height 8, slant height is √((6 − 4)² + 8²) = √68 ≈ 8.25. Then lateral area is π(4 + 6) × 8.25 ≈ 259.4 square units.

Why does the volume formula include the term r × R?

The volume formula V = (1/3)πh(r² + rR + R²) comes from integrating circular cross-sections at each height. This term accounts for the interaction between the two radii across the frustum's length. If you set r = R (cylinder), it simplifies to (1/3)πh × 3R² = πR²h, confirming the result. The mixed term ensures smooth transitions between any pair of radii.

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