Truncated Cone Calculator - Volume, Slant, and Area

A truncated cone calculator is a tool that takes the larger base radius R, the smaller top radius r, and the perpendicular height h of a right circular frustum and returns every common property in real time. Enter R, r, and h, and the page reports the volume, slant height, lateral area, both base areas, and total surface area together.

• • Geometry homework: Solve problems that give you R, r, and h and ask for V, A_L, or A without re-deriving the frustum formula each time.
• • Planters and lampshades: Recover the volume or surface area of a real frustum-shaped object (flowerpot, lampshade, tapered bucket) when only the radii and height are easy to measure.
• • Tanks and hoppers: Estimate the capacity or material needed to coat or paint a conical transition, hopper bottom, or funnel that flares from a smaller top to a larger base.

Geometry problems and real-world parts rarely give you the volume directly; the easy-to-measure quantities are the two radii and the height. The calculator accepts exactly those three numbers and returns every other property that follows from them, assuming a right circular frustum where the two bases sit parallel to each other and the slanted side is rotationally symmetric.

For users who only need the capacity of the same shape with built-in unit switching, the Truncated Cone Volume Calculator applies the same V = pi h (R squared + R r + r squared) divided by 3 identity and reports cm cubed, m cubed, in cubed, L, and US gal in real time.

The calculator applies four closed-form formulas from the geometry of a right circular frustum. All four are exact, not approximations, and they are valid for any non-negative R, r, and h including the cylinder limit R equals r and the full-cone limit r equals 0.

• R: Larger (lower) base radius of the frustum, in the same unit as r and h.
• r: Smaller (upper) top radius of the frustum, in the same unit as R and h.
• h: Perpendicular height between the two parallel bases, in the same unit as R and r.
• s: Slant height along the slanted side, equal to sqrt((R - r) squared + h squared).

All four formulas use Math.PI internally, so the volume is correct well beyond the four-decimal display and the area values are accurate to the third decimal place. Keep all three inputs in the same unit family, since mixed units (for example R in cm and h in inches) produce silently wrong results.

According to Wolfram MathWorld, V = (pi h / 3) (R squared + R r + r squared) and s = sqrt(h squared + (R - r) squared) for a right circular frustum

When the focus is the surface area and slant height only, the Frustum Cone Area applies A = pi (R squared + r squared + s (R + r)) directly and returns the lateral, base, and total areas without re-deriving the volume.

Four short ideas explain why a frustum is special, how the slant height closes the geometry, and why the cylinder and full cone fall out as limiting cases.

The four output properties (volume, slant height, lateral area, total area) all flow from R, r, and h, so once you know those three, every other common property of the frustum follows without additional measurement.

Setting the smaller top radius to zero reduces the frustum to a full right cone, and the Cone Volume Calculator applies V = (pi h / 3) R squared to that limiting case in real time.

Four short steps take you from the three defining dimensions of a frustum to the volume, slant height, and every surface area you might need.

1. 1 Enter the larger base radius R: Type the radius of the lower base. Use the same unit as r and h, and use the radius (half the diameter) rather than the diameter.
2. 2 Enter the smaller top radius r: Type the radius of the upper base. Set r to 0 for a full right cone, or set r equal to R for a right cylinder.
3. 3 Enter the perpendicular height h: Type the height measured perpendicular to the bases, not the slant side. The slant height is derived separately.
4. 4 Read the result panel: The primary volume appears at the top to four decimal places. The slant height, lateral area, both base areas, and total surface area update at the same time for cross-checking.

For a tapered flowerpot with R = 5 cm, r = 3 cm, and h = 6 cm, the calculator reports V approx 307.876 cm cubed, s approx 6.325 cm, A_L approx 158.953 cm squared, and A approx 265.768 cm squared, giving both the soil capacity and the glaze area in one pass.

When the frustum is just the lateral area of a full cone and the upper radius is zero, the Lateral Area of Cone Calculator applies A_L = pi r l directly and returns the matching slant height and total surface area from the same radius and height.

Five practical reasons to use a dedicated truncated cone calculator instead of juggling the frustum identities by hand.

• • One input set, every output: R, r, and h are the only numbers you need. The calculator returns volume, slant height, lateral area, both base areas, and total surface area together.
• • Closed-form formulas: Every result uses the exact frustum identities (V, A_L, A), with no numerical integration or approximation hidden behind the scenes.
• • Limiting cases included: R equals r recovers a cylinder, r equals 0 recovers a full cone, and h equals 0 reduces the frustum to two flat circles, so no separate tool is needed for these shapes.
• • Educational reference: The formula box, worked examples, and concept cards explain where each identity comes from, doubling as a quick reference for students and a sanity check for engineers.
• • Design and material use: The lateral area and total area give the surface area of paint, glaze, fabric, or sheet metal needed to cover a frustum-shaped part.

These benefits show up most clearly when the inputs on a part or plan are not the outputs you need. A pot label that lists only R, r, and h, or a homework problem that asks for V, A_L, and A, both flow through the same tool.

If the spec only lists a diameter instead of a radius, the Diameter of Cone Calculator recovers d from height, slant height, volume, lateral area, or base area in the same four-decimal display used here.

Three factors control the precision of the truncated cone results, plus three important limitations to keep in mind when interpreting the volume, slant height, and area you read back.

• • This calculator assumes a true right circular frustum. It does not handle oblique frustums, non-circular bases, or shapes with non-circular tops.
• • It accepts only R, r, h at a time. If you have measured the frustum by slant height and one radius, recover the missing dimension with a separate tool first.
• • It is not a measurement tool. Real frustums still need a ruler, caliper, or tape; this tool only does the arithmetic from the dimensions you already have.

For real-world frustums that are slightly out of round (cast concrete hoppers, hand-thrown clay planters, 3D-printed parts), measuring R and r as the average radius of each end still gives a reliable estimate.

According to Omni Calculator, the truncated cone uses R, r, and h to return volume, surface areas, and slant height in real time.

According to Wikipedia (Frustum), the lateral area is pi s (R + r) and the total area is pi (R squared + r squared + s (R + r)).

When the truncated cone in question is the conical end of a cylindrical tank or pipe, the Cylinder Volume Calculator gives the matching cylinder volume from the same base radius so the two parts can be combined into a total capacity.