Conical Frustum Calculator

Calculate volume, total and lateral surface area, slant height, and circular end areas for a right circular conical frustum from its two radii and perpendicular height.

Frustum dimensions

Circular measurement type Choose how both circular ends are measured.

Larger radius (R) The larger circular end.

Smaller radius (r) May be zero for the cone limit.

Perpendicular height (h) The shortest distance between the parallel circular ends—not the slanted side.

Length unit

Decimal places

π value

Your measurements stay in this browser. Use one common unit for all three lengths; the unit selector labels results but does not convert mixed inputs.

Results

Enter the two circular measurements and perpendicular height.

Your volume, surface areas, slant height, and formula substitutions will appear here.

Conical Frustum Formulas

A conical frustum is the part left when the top of a cone is cut by a plane parallel to its circular base. This calculator assumes a right circular frustum: the centres of both circular ends lie on the same perpendicular axis.

Measurement	Formula	What it includes
Slant height	`s = √(h² + (R − r)²)`	Straight side distance between the circular rims
Volume	`V = (πh/3)(R² + Rr + r²)`	Space inside the frustum
Lateral area	`L = π(R + r)s`	Curved side only
Total surface area	`S = L + πR² + πr²`	Curved side plus both circular ends
Large base area	`A_R = πR²`	Larger circular end
Small base area	`A_r = πr²`	Smaller circular end

Units: R, r, h, and s are lengths. Areas use squared units and volume uses cubed units. Displayed values are rounded only after the calculator completes the formulas at full JavaScript floating-point precision.

Worked Example

Bucket-shaped frustum

For R = 6 cm, r = 3 cm, and h = 4 cm, slant height is 5 cm. Volume is 84π ≈ 263.89 cm³.

Surface material

The same frustum has lateral area 45π ≈ 141.37 cm² and total closed surface area 90π ≈ 282.74 cm².

How to Measure a Frustum Correctly

Measure each circular end across its centre. Divide by two if you measured diameter, or choose Diameter in the calculator.
Use the larger value for R and the smaller value for r.
Measure h perpendicular to the bases. Do not substitute the slanted side length.
Keep all three measurements in one unit before calculating.

Physical containers may have wall thickness, rounded rims, seams, or non-circular ends. The result models ideal geometry and should not replace detailed engineering measurements.

Special Cases and Checks

If r = 0, the frustum becomes a cone and the formulas reduce to the standard cone formulas.
If R = r, the sides are vertical, s = h, and the shape becomes a cylinder.
The calculator requires R ≥ r ≥ 0, R > 0, and h > 0.
Inputs from 10⁻¹⁰⁰ through 10¹⁰⁰ are supported for positive measurements. Values that overflow safe browser arithmetic are rejected clearly.

Methodology and Assumptions

The page uses the standard right-circular-frustum identities shown above. Slant height follows from the Pythagorean theorem applied to perpendicular height h and horizontal offset R − r. All calculations run locally; changing the π option is useful when a classroom problem specifies an approximation.

Reference: formulas cross-checked with Wolfram MathWorld: Conical Frustum. Last reviewed: July 30, 2026.

Conical Frustum Calculator FAQs

What is the volume formula for a conical frustum?

For larger radius R, smaller radius r, and perpendicular height h, volume is V = (πh/3)(R² + Rr + r²).

How do I find the slant height of a frustum?

For a right circular conical frustum, slant height is s = √(h² + (R − r)²).

What is the total surface area of a conical frustum?

Total surface area is π(R + r)s + πR² + πr². This includes the curved side and both circular ends.

What is lateral surface area?

Lateral surface area is the curved side only. For a conical frustum it is L = π(R + r)s.

Can I enter diameters instead of radii?

Yes. Choose Diameter before calculating. The calculator divides both entered diameters by two and then applies the radius-based formulas.

Does the order of the radii matter?

The formulas are symmetric, but this calculator labels R as the larger radius and r as the smaller radius to keep the diagram and substitutions clear.

Can the smaller radius be zero?

Yes. A zero smaller radius is the limiting case of a full cone, so the standard cone formulas are recovered.

Does the calculator convert between units?

No. Choose one unit and enter every length in that same unit. The calculator labels areas with squared units and volume with cubed units.

Are my measurements uploaded?

No. The calculation runs locally in your browser and the page does not send your measurement values to a server.