Truncated Cone Volume Calculator

How to use the truncated cone volume calculator

What this calculator does

This calculator finds the volume of a right circular truncated cone, also called a conical frustum. It uses the top radius, base radius, and perpendicular height in the standard identity V = πh(r² + rR + R²) / 3. It also links vertical height and slant height through the right-triangle relation s² = h² + (R – r)². The result is an exact geometric estimate for an ideal frustum; it does not account for wall thickness, rounded corners, manufacturing tolerances, irregular profiles, or material displacement.

When to use it

Use the calculator when estimating the capacity of a tapered container, checking a geometry homework problem, sizing a frustum-shaped hopper or planter, or comparing a design change in height or radius. It is also useful when a drawing gives slant height instead of vertical height, because the calculator automatically derives the missing dimension before applying the volume formula.

How to calculate

1. The calculator opens with a complete demonstration: top radius 1 cm, base radius 2 cm, height 5 cm, and the corresponding slant height. The live results and a validated example XLSX are available immediately.
2. Select the Length unit you want to use. A unit change converts the existing dimensions so the physical frustum remains the same.
3. Replace Top radius (r) and Base radius (R) with your positive measurements.
4. Edit either Height (h) or Slant height (s). The field you edit most recently becomes the driver, and the other dimension is recalculated.
5. Read Volume and the supporting areas, review the geometry table, then use Download Excel to export the current unrounded model.
6. Reset clears the demonstration and all calculated content. Download Excel is then disabled until you enter a complete valid set again.

Input guide

Length unit is required and accepts millimeters, centimeters, meters, inches, or feet. It controls every length label, square-unit result, cubic-unit result, and workbook value. For example, choosing centimeters keeps the demonstration at 1 cm, 2 cm, and 5 cm. A common mistake is to change a unit label without converting the numbers; this calculator converts them automatically.

Top radius (r) is a required positive decimal in the selected unit. Enter the radius, not the diameter. A realistic example is 1 cm. Increasing it while other dimensions stay fixed increases volume, top area, average cross-section, and usually decreases the radius ratio. Grouping commas are accepted only in conventional forms such as 1,250.5; decimal commas and scientific notation are rejected to avoid ambiguous interpretation.

Base radius (R) is also required and positive. A realistic example is 2 cm. The formula is symmetric in the two radii, so the calculator remains mathematically valid if the top radius is larger than the base radius, although many practical vessels use a wider base. Do not enter circumference or diameter.

Height (h) is the perpendicular distance between the parallel circles. It is required when height is the active driver and must be positive. The demonstration uses 5 cm. Increasing height increases volume in direct proportion. Editing height recalculates slant height, so you should not try to preserve an old slant measurement after changing height or either radius.

Slant height (s) is the straight distance along the side. It must be positive and cannot be less than |R – r|; otherwise no real vertical height exists. The demonstration value is about 5.099 cm. Editing slant height makes it the active driver and recalculates height. The common interpretation error is to substitute slant height directly into the volume formula; volume requires perpendicular height.

Output guide

Volume is the primary exact geometric result, displayed in the cube of the selected length unit. Volume in liters is a capacity conversion derived from cubic meters; NIST explains that 1 liter equals 1 cubic decimeter and 1 milliliter equals 1 cubic centimeter in its guidance on SI units for volume. A zero volume cannot occur with valid positive inputs, while very large values should prompt a unit check.

Computed height and Computed slant height show the internally consistent pair used by the model. Top face area and Base face area are the circular end areas. Average cross-section equals volume divided by height; it is a derived average area, not the simple arithmetic mean of the two end areas. The header pills report the selected unit, current height, radius ratio, and whether height or slant height is driving the calculation. The geometry table lists every dimension and derived quantity with its symbol, value, unit, and role.

Worked example

With r = 1 cm, R = 2 cm, and h = 5 cm, the radius expression is 1² + 1×2 + 2² = 7. Therefore V = π×5×7/3 = 36.6519 cm³, displayed on first open. The matching slant height is √(5² + (2 – 1)²) = √26 = 5.0990 cm. For a deeper derivation and related surface formulas, see Wolfram MathWorld's conical frustum reference.