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
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
AR = πR²
Larger circular end
Small base area
Ar = 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.
