Surface of Revolution Area Calculator
When a straight line segment is revolved a full turn about an axis, it sweeps out the lateral surface of a cone or a frustum (a truncated cone). The area of that surface follows directly from Pappus's centroid theorem: the arc length of the generating segment times the circumference traced by its centroid. Enter the radius at each end of the segment and the axial span between them. The calculator returns the slant length, the centroid radius, and the lateral surface area.
Surface of revolution formula
Slant length L = sqrt((r2 - r1)^2 + h^2)
Centroid radius R = (r1 + r2) / 2
Surface area A = 2 * pi * R * L = pi * (r1 + r2) * L
This is Pappus's first centroid theorem applied to a straight segment: lateral area equals arc length L times the circumference 2 * pi * R traced by the centroid. The centroid of a uniform straight segment lies at its midpoint, so R is the average of the two radii.
Notes on use
- Setting r1 equal to r2 gives a cylinder side wall: A = 2 * pi * r * h.
- Setting one radius to 0 gives a full cone: A = pi * r * slant length.
- The generating segment must not cross the axis; both radii are distances and must be 0 or positive.
- All inputs use the same length unit; the area is returned in that unit squared.
Surface of revolution: frequently asked questions
What is a surface of revolution?
A surface of revolution is the surface created when a plane curve is rotated a full turn about an axis lying in the same plane. Revolving a line segment about an axis produces the lateral (side) surface of a cone or a truncated cone (frustum). Revolving a curve produces more general curved surfaces.
What is Pappus's theorem for surface area?
Pappus's first centroid theorem states that the lateral surface area equals the arc length of the generating curve multiplied by the distance the curve's centroid travels: A = L times 2 times pi times R, where L is the arc length and R is the distance from the axis to the centroid of the curve.
Does this calculator handle a straight line segment?
Yes. This calculator revolves a straight segment between two points whose distances from the axis are r1 and r2 and whose span along the axis is h. The arc length is the straight-line distance, and the centroid sits at the midpoint, giving the standard frustum lateral area formula A = pi times (r1 + r2) times slant length.
What is the slant length?
The slant length is the straight-line distance along the segment being revolved. For a segment whose radii change from r1 to r2 over an axial span h, the slant length equals the square root of ((r2 minus r1) squared plus h squared). It is the hypotenuse of the radial change and the axial span.
Sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov (geometry and integral reference).
- Formula derived from Pappus's centroid theorem (standard result of integral calculus).
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.