About

A frustum (or frustum of a cone) is the solid remaining after a smaller cone is cut from a larger cone by a plane parallel to its base. Miscalculating its surface area leads to material waste in sheet-metal fabrication, incorrect paint estimates on tapered tanks, and errors in HVAC duct sizing. This calculator computes lateral surface area using A_lat = π(R + r)l, where l is the slant height. If only the perpendicular height h is known, slant height is derived via the Pythagorean relation l = √h² + (R − r)². The tool assumes a right circular frustum with parallel circular bases.

Precision matters. A 5% error in the larger radius R compounds quadratically in the base area and linearly in the lateral area. The calculator also outputs volume using the prismoidal formula, useful for capacity estimation of tapered vessels. All results assume Euclidean geometry. Pro tip: for real sheet-metal work, add 2 - 5% to the lateral area for seam and overlap allowance.

Formulas

The lateral surface area of a frustum is derived by integrating the circumference of infinitesimal rings along the slant surface. The closed-form result is:

A_lat = π(R + r)l

When the slant height l is not directly measured, it is computed from perpendicular height h and the radii difference:

l = √h² + (R − r)²

The total surface area sums the lateral area with both circular bases:

A_total = πR² + πr² + π(R + r)l

Volume follows the prismoidal (frustum) formula, exact for second-degree surfaces:

V = πh3(R² + Rr + r²)

Where: R = radius of the larger base, r = radius of the smaller (top) base, h = perpendicular height between the two bases, l = slant height along the lateral surface, V = enclosed volume. All lengths must share the same unit.

Reference Data

Property	Formula	Units	Typical Use
Lateral (Side) Surface Area	π(R + r)l	m²	Sheet metal, painting, insulation
Top Base Area	πr²	m²	Cap/lid sizing
Bottom Base Area	πR²	m²	Foundation footprint
Total Surface Area	π(R² + r² + (R + r)l)	m²	Total material estimate
Slant Height	√h² + (R − r)²	m	Derived when not measured directly
Volume	πh3(R² + Rr + r²)	m³	Capacity, fill volume
Semi-vertical Angle	arctan(R − rh)	°	Taper specification
Mean Radius	R + r2	m	Approximation in thin-wall analysis
Centroid Height (from base)	h(R² + 2Rr + 3r²)4(R² + Rr + r²)	m	Structural analysis, center of mass
Ratio Check (r = 0)	Degenerates to full cone	-	Validation edge case
Ratio Check (r = R)	Degenerates to cylinder	-	Validation edge case
Apex Distance (full cone height)	RhR − r	m	Pattern development for sheet metal
Unrolled Arc Angle	360R√H² + R²°	°	Flat pattern layout
Taper per Unit Length	R − rh	m/m	CNC machining specification

Frequently Asked Questions

When r = 0, the frustum degenerates into a full cone. The lateral area formula simplifies to A_lat = πRl, which is the standard cone lateral area. The calculator handles this edge case correctly.

The perpendicular height h is measured vertically between the two parallel bases. The slant height l is the distance along the lateral surface from one base edge to the other. They are related by l = √h² + (R − r)². Slant height is always greater than or equal to perpendicular height. They are equal only when R = r (a cylinder).

Mathematically, the formulas are symmetric with respect to which base is larger. However, by convention R denotes the larger base. If you enter a top radius exceeding the bottom radius, the calculator still produces correct results because the difference (R − r) is squared in the slant height computation, eliminating sign issues.

The lateral surface unrolls into an annular sector. The theoretical area is π(R + r)l, but the bounding rectangle of the unrolled pattern is larger. Depending on taper angle, expect 15 - 40% scrap. For precise nesting, compute the apex distance H = Rh ÷ (R − r) and the arc angle to lay out the flat pattern.

The frustum volume equals the full cone volume minus the removed tip cone volume. If the full cone has height H = Rh ÷ (R − r) and the removed tip has height H − h, then V_frustum = π3(R²H − r²(H − h)). This simplifies to the prismoidal formula.

Yes. Metals expand linearly with temperature. Steel has a coefficient of approximately 12 × 10⁻⁶ K⁻¹. A 100°C rise increases linear dimensions by about 0.12%, and surface area by roughly 0.24%. For most fabrication, this is negligible, but for precision vessels or cryogenic applications, apply the expansion factor to each radius and height before calculating.