BYTETOOLS

Cone Frustum Calculator

Calculate the volume, slant height and surface area of a truncated cone, and get the flat pattern radii and included angle for sheet metal or papercraft.

205.2507
Volume (cm³)
4.4721
Slant height (cm)
112.397
Lateral area (cm²)
219.2112
Total area (cm²)

Working

  • Volume = (πh/3)(R² + Rr + r²) = 205.25072 cm³
  • Slant height l = √(h² + (R - r)²) = 4.472136 cm
  • Lateral area = π(R + r)l = 112.397036 cm²
  • Bottom circle area = πR² = 78.539816 cm²
  • Top circle area = πr² = 28.274334 cm²
  • Total surface area = 219.211186 cm²
  • Liters (L)0.2053
  • US gallons0.0542
  • Imperial gallons0.0451
  • Cubic meters (m³)0.0002
  • Cubic feet (ft³)0.0072

Flat pattern (development)

  • Outer radius = 11.18034 cm
  • Inner radius = 6.708204 cm
  • Included angle = 160.9969°
  • Radial width (equals the slant height) = 4.472136 cm

Cut this annular sector from flat stock and roll it up to make the frustum. Add a seam allowance; the pattern is the finished size.

What is the Cone Frustum Calculator?

The ByteTools Cone Frustum Calculator solves a truncated cone from its two radii and vertical height.

  • Volume, slant height, lateral area, both end areas and total surface area
  • Flat pattern development with outer radius, inner radius and included angle, drawn to scale
  • Accepts radii or diameters in metres, centimetres, millimetres, feet or inches
  • Handles the full-cone case when one radius is zero and the cylinder case when both are equal
  • Volume converted to litres, gallons, cubic metres and cubic feet
  • Runs entirely in your browser — nothing leaves your device

How to use the Cone Frustum Calculator

  1. 1

    Choose whether you are entering radii or diameters, and pick your measurement unit.

  2. 2

    Enter the bottom size, the top size and the vertical height.

  3. 3

    Read the volume, slant height, lateral area and total area in the summary tiles.

  4. 4

    Check the working panel for the formulas and the volume conversions.

  5. 5

    Use the flat pattern panel for the outer radius, inner radius and included angle, then copy the results.

About the Cone Frustum Calculator

The ByteTools Cone Frustum Calculator solves a truncated cone from its two radii and vertical height. It returns the volume (πh/3)(R² + Rr + r²), the slant height √(h² + (R - r)²), the lateral surface π(R + r)l, both circular end areas and the total surface area, with the working shown line by line.

The part makers actually need is the flat pattern: the shape you cut from sheet stock and roll up. The tool gives the outer radius, the inner radius and the included angle of the annular sector, and draws it to scale. When the two radii are equal it says so and gives you a rectangle instead, because a cylinder does not unroll into a sector.

Volumes are converted to litres, gallons, cubic metres and cubic feet for hopper and container work. Everything is computed in your browser, so nothing is uploaded and the page works offline in a workshop.

Frequently asked questions

What is the volume formula for a cone frustum?

V = (πh/3)(R² + Rr + r²), where R and r are the two radii and h is the vertical height. With R = 5, r = 3 and h = 4 that gives 196π/3, about 205.25 cubic units. Setting r to zero reduces it to the familiar cone formula πr²h/3.

How do you find the slant height of a frustum?

Use the Pythagorean theorem on the height and the difference between the radii: l = √(h² + (R - r)²). For R = 5, r = 3 and h = 4 the slant height is √(16 + 4) ≈ 4.472. Note it uses the difference of the radii, not their sum.

How do I make a flat pattern for a cone frustum?

Cut an annular sector. Its outer radius is l·R/(R - r), its inner radius is l·r/(R - r), and its included angle is 2πR divided by that outer radius. For R = 5, r = 3, h = 4 the pattern has radii of 11.18 and 6.708 with a 161° angle.

What is the lateral surface area of a frustum?

It is π(R + r)l, the area of the sloping side without either flat end. For R = 5, r = 3 and slant 4.472 that is about 112.4 square units. Use this figure for material, paint or wrap; add πR² and πr² if you also need the two ends.

Why does the calculator give a rectangle instead of a sector?

Because when both radii are equal the shape is a cylinder, not a frustum, and its side unrolls into a plain rectangle whose width is the circumference and whose height is the vertical height. The sector formulas would divide by zero, so that case is handled separately.

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