The Conical Frustum Calculator converts two parallel circular radii and the perpendicular height between them into volume, slant height, lateral surface area and total surface area. A conical frustum is a right circular cone with the tip cut off by a plane parallel to the base. The page redraws a labelled figure with both radii and height marked.
Enter larger radius R, smaller radius r and height h. Related pages cover the full Cone Calculator before the cut, the Pyramid Frustum Calculator for square truncated pyramids, and the Cylinder Calculator when R equals r.
Calculate the volume of a conical frustum
Volume of a conical frustum mixes both base areas with a geometric-mean cross term under a one-third factor. The Conical Frustum Calculator uses the standard truncated-cone formula. Radii and height must be positive.
V = (pi * h / 3) * (R^2 + R * r + r^2)When r approaches 0 the formula reduces to (1/3) * pi * R^2 * h, the full cone. When R equals r it reduces to pi * R^2 * h, the cylinder. Those limits are useful sanity checks.
For R = 5, r = 3, h = 6:
V = (pi * 6 / 3) * (25 + 15 + 9) = 2 * pi * 49 = 98 * pi approx 307.88Exact form 98 * pi matches the outline fixture. Volume is not the average of the two cylinder volumes at R and r times height; that average would omit the R*r term and under-count. The cross term matters.
Calculate the slant height and surface area
Slant height s is the distance along the slanted side between the two circular edges. For a right frustum, s is the hypotenuse of height h and radial difference R - r. Lateral area uses pi * (R + r) * s. Total surface area adds both circular bases.
s = sqrt(h^2 + (R - r)^2)
Lateral SA = pi * (R + r) * s
Total SA = Lateral SA + pi * R^2 + pi * r^2Using perpendicular height in place of slant under-counts the wall. Open buckets that omit one base subtract that circle from total area.
For R = 5, r = 3, h = 6:
s = sqrt(36 + 4) = sqrt(40) = 2 * sqrt(10) approx 6.32
Lateral SA = pi * (5 + 3) * sqrt(40) = 8 * pi * sqrt(10) approx 79.48
Total SA = Lateral SA + pi * 25 + pi * 9 = Lateral SA + 34 * pi approx 186.30Solve a frustum with radii 5 and 3, height 6
Take R = 5, r = 3 and h = 6 as the reference conical frustum. Volume is 98 * pi. Slant height is sqrt(40). 1. Volume
V = 98 * pi approx 307.882. Slant height
s = sqrt(40) approx 6.323. Lateral surface area
Lateral SA = 8 * pi * sqrt(10) approx 79.48A full cone that continues to a point beyond the small end would have height larger than 6; recovering that cone needs similar triangles and is optional for volume of the frustum alone. This page computes the truncated solid directly.
Similar triangles relate the missing tip to the kept frustum. If the large base radius is R and the small is r, the scale factor from the full cone to the removed tip is r/R. Full cone height H satisfies r/R = (H - h)/H, so H = h / (1 - r/R). For R = 5, r = 3, h = 6: H = 6 / (1 - 3/5) = 6 / (2/5) = 15. Full cone volume is (1/3)*pi*25*15 = 125 * pi; tip volume is (1/3)*pi*9*9 = 27 * pi; difference 98 * pi matches the frustum formula.
Recognise frustums in everyday objects
Truncated cones appear when a full cone is cut by a plane parallel to the base. Buckets, lampshades, drinking glasses that taper, planter pots and some funnels approximate this shape. Measure both opening radii (or diameters) and the vertical height between the parallel top and bottom for best accuracy.
Tapered ducts with square ends belong on the Pyramid Frustum Calculator instead. Pipes with constant radius belong on the Cylinder or Tube calculators. A shade that flares with a curved profile is not a right conical frustum; the formulas here assume straight generators between the two circles.
Build a lateral net with slant height
Lateral surface area uses slant height s = sqrt(h^2 + (R - r)^2), not perpendicular height alone. For R = 5, r = 3, h = 6, s = sqrt(40).
The wall unrolls to a sector-like frustum net whose straight edges have length s and whose arcs match the two circumferences 2 * pi * R and 2 * pi * r.
Lateral SA = pi * (R + r) * s = 8 * pi * sqrt(10) for the reference case.
Closed solids add both bases: pi * R^2 + pi * r^2 = 34 * pi here. Open buckets that omit the large base subtract pi * R^2; omit the small opening by subtracting pi * r^2. Using h in place of s under-counts any coating on the slanted wall.
Unit note: both radii and height share one length unit. Volume is in cube units; slant and surface areas use that same length for square measure. Enter diameters only after halving, or use diameter fields if the tool halves them before the formulas run.
Frequently asked questions
What is conical frustum volume?
Volume is V = (pi * h / 3) * (R^2 + R*r + r^2). For R = 5, r = 3, h = 6, V = 98 * pi approx 307.88.
What is slant height?
Slant height is s = sqrt(h^2 + (R - r)^2). For the reference frustum, s = sqrt(40) approx 6.32.
What is lateral surface area?
Lateral surface area is pi * (R + r) * s. It uses slant height, not perpendicular height alone.
How does a frustum relate to a cone?
A conical frustum is a cone with the tip removed parallel to the base. Use the Cone Calculator when the tip is still present.
What if both radii are equal?
The solid is a cylinder. Volume becomes pi * R^2 * h and lateral area becomes 2 * pi * R * h.
Can the Conical Frustum Calculator use diameters?
Yes. Halve each diameter to a radius before applying the formulas.
What units does the Conical Frustum Calculator use?
Radii and height share one length unit. Volume uses cube units; surface areas use square units.
Is a bucket always a conical frustum?
Many buckets approximate one. Confirm that top and bottom are circular and parallel and that the side is straight along a generator.
Can volume be full cone minus tip cone?
Yes, when the cut is parallel to the base. For the reference numbers, 125 * pi - 27 * pi = 98 * pi, matching the direct frustum formula.
What if I measure diameters instead of radii?
Halve each diameter. The volume formula is homogeneous in the radii, so consistent halving preserves the result.
Does the Conical Frustum Calculator need the apex height?
No. Enter the two radii and the frustum height h. Apex height is optional for similar-triangle checks only.
What is the reference frustum volume?
For R = 5, r = 3, h = 6, volume is 98 * pi approx 307.88 with slant sqrt(40).
What is the matching full cone height?
By similar triangles, H = 15 for the reference frustum. Full cone volume 125 * pi minus tip 27 * pi equals 98 * pi.
Summary
Conical Frustum Calculator returns volume, slant height and surface areas for a truncated right circular cone from two radii and height, with substituted arithmetic and a labelled figure. Worked numbers match R = 5, r = 3, h = 6: V = 98 * pi and slant sqrt(40).
Use the Cone Calculator for the uncut solid and the Pyramid Frustum Calculator for square truncated pyramids.