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How to use
A cone frustum is the portion of a cone between two parallel circular bases. From radii R and r plus perpendicular height h, this calculator returns volume, slant height, lateral area, both base areas and total surface area. For frustum radii 5 cm and 2 cm with perpendicular height 4 cm, the slant height is √(4²+(5−2)²)=5 cm. Volume is 52π ≈ 163.3628 cm³ and total surface area is 64π ≈ 201.0619 cm². If you measured diameters, halve both before entering the radii.
- Enter the first base radius R.
- Enter the second radius r and perpendicular height h using the same length unit.
- Calculate the geometry outputs.
How it is calculated
Example
Example: R=5, r=3, h=8 gives ℓ=√68≈8.246 and V=π×8×(25+15+9)/3≈410.50 cubic units.
Important notes
Use the same length unit for all dimensions. Areas are squared units and volume is cubed units. If one radius is zero, the shape reduces to a cone; if the radii are equal, it reduces to a cylinder.
Worked examples and interpreting results
A right conical frustum with base radii 3 and 1 cm and height 4 cm has volume 52π/3 cm³. The formula combines both radii; replacing them with their average in a cylinder formula is incorrect.
| Case | Calculation | Result |
|---|---|---|
| Volume | π × 4 × (3² + 3×1 + 1²) ÷ 3 | 54.454273 cm³ |
| Slant height | √(4² + (3−1)²) | 4.472136 cm |
| Volume: R=5 cm, r=2 cm, h=4 cm | π × 4 × (25 + 10 + 4) ÷ 3 | 163.362818 cm³ |
| Total area: R=5 cm, r=2 cm, slant height 5 cm | π × (5 + 2) × 5 + π × (25 + 4) | 201.06193 cm² |
How to check the result
Equal radii produce a cylinder; making one radius zero produces a full cone. Enter radii rather than diameters and use perpendicular height, not slant height, for volume.
Frustum of a cone: volume, slant height and surface area
A right circular frustum is specified by large radius R, small radius r and perpendicular height h. Its volume is πh(R²+Rr+r²)/3; slant height is √(h²+(R−r)²).
- Volume for R=4, r=2, h=3: π × 3 × (16 + 8 + 4) ÷ 3 = 28π units³
- Slant height with the same dimensions: √(3² + (4−2)²) = √13 units
- Total area with the same dimensions: π×(4+2)×√13 + π×(16+4) = π(6√13 + 20) units²
Important distinction
Slant height is not the vertical height used by the volume formula. Total surface area includes the two circular bases; lateral area alone does not.
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Frequently asked questions
Must R be larger than r?
No. The formulas are symmetric in the two radii, so either order works.
What is the difference between h and ℓ?
h is perpendicular height; ℓ is the slant distance along the side.
Does the calculator include both bases?
Yes. Total surface area includes lateral area plus both circular bases.
How is the slant height calculated correctly?
For a right circular frustum, s=√(h²+(R−r)²), and lateral area is π(R+r)s. An oblique cone or a non-circular frustum requires a different geometric model.
Does the frustum formula work when both radii are equal?
Yes. The shape becomes a cylinder: if R=r, volume is πR²h and slant height is h. Setting one radius to zero gives a right circular cone.