How this instrument works
This combined page solves a truncated cone (frustum) completely from its two radii and height, returning both the enclosed volume and the complete exterior surface area together. The volume follows the standard frustum formula, V = πh(R²+Rr+r²) ⁄ 3. The surface area adds the slanted lateral surface, found via the slant height √(h²+(R−r)²), to both flat circular caps: total = π(R+r)·slant + πR² + πr².
This goes further than this site's Truncated Cone Volume page, which returns only the enclosed volume. Here, both figures come back together, useful whenever a frustum-shaped object — a bucket, a lampshade, a hopper — needs both a capacity estimate and a material or coating estimate solved from the same three measurements.
Both formulas correctly collapse to their simpler cylinder or cone equivalents at the right extremes: setting the two radii equal turns the frustum into a plain cylinder, and shrinking the top radius to zero turns it into an ordinary full cone, with both the volume and the surface area formulas reducing exactly to those simpler shapes' own known equations.
- Enter the larger, bottom radius into the Bottom radius field.
- Enter the smaller, top radius into the Top radius field.
- Enter the height into the Height field.
- Read Volume and Total surface area: the sheet computes both simultaneously, finding the slant height internally.
Worked example — radii 5 and 3, height 4
A frustum has bottom radius 5, top radius 3, and height 4. Its volume is about 205.25, and its total surface area — the slanted side plus both flat circular caps — is about 219.21, both figures returned together from the same three measurements, with the slant height (√20≈4.47) computed internally along the way.
Equal radii (5 and 5) turn the frustum into a plain cylinder: volume 100π and surface area 90π, matching the ordinary cylinder formulas exactly. A top radius of 0 instead turns it into a full cone, with both formulas reducing to their standard cone equivalents.
Questions
How is this different from the Truncated Cone Volume page?
That page returns only the enclosed volume. This page returns BOTH the volume and the complete exterior surface area together, for whenever both a capacity figure and a material estimate are needed for the same frustum.
What does the surface area include?
The full exterior: the slanted lateral surface running between the two circular edges, plus BOTH flat circular caps — the smaller top circle and the larger bottom circle.
What happens when the two radii are equal?
The frustum's slanted side straightens out entirely, turning it into a plain cylinder — both the volume and surface area formulas correctly reduce to the ordinary cylinder equivalents, πr²h and 2πrh+2πr².
What happens when the top radius is zero?
The frustum becomes an ordinary full cone, with the smaller circle shrunk to a single point — both formulas reduce exactly to the standard cone volume and surface area equations.
Is the slant height needed as a separate input?
No — it's computed internally from the height and the difference between the two radii, via the Pythagorean theorem, so only the two radii and the height need entering directly.