Right Circular Cone Calc: find A, V, A_L, A_B.

How to use this right circular cone calculator

What this calculator does

This calculator solves a right circular cone from any two of its three defining lengths: base radius, vertical height, and slant height. It then returns the total surface area, enclosed volume, curved lateral surface area, circular base area, and the missing length. The model assumes the apex is directly above the center of a circular base. It does not apply to an oblique cone, a truncated cone, or a shape with an elliptical base.

When to use it

Use it to check geometry homework, estimate sheet material for a cone-shaped cover, compare container capacity, or convert a measured cone between metric and U.S. customary units. It is also useful when the easiest physical measurement is the slant height rather than the perpendicular height.

How to calculate

1. The calculator opens with a demonstration: a 3 cm base radius and 4 cm vertical height. Those values immediately produce a complete result and a validated Excel workbook.
2. Replace the demonstration by entering any two positive measurements. Leave the third value blank so the calculator knows which length to solve. You may enter all three only when they are mutually consistent.
3. Choose the unit beside each entered measurement. Changing a measurement unit converts the current numeric value rather than merely relabeling it. Choose the Result unit separately to control the length, square-unit, and cubic-unit output labels.
4. Read Total surface area (A) as the primary result, then review Volume (V), Lateral surface area (Aₗ), Base area (Aᵦ), and the solved length. The detail table shows the same values beside their formulas.
5. Select Download Excel to export the current validated state. Reset clears every measurement and result; after Reset, the Excel button remains unavailable until two complete, compatible values are entered again.

Input guide

Base radius value is a required positive decimal whenever radius is one of the two known measurements. It is the distance from the base center to its edge. Enter plain U.S.-style decimals such as 3 or 12.5; a comma is accepted only in a valid thousands group, and scientific notation is rejected. A larger radius increases every area and increases volume quadratically. Do not enter the diameter.

Base radius unit specifies millimeters, centimeters, meters, inches, or feet for the radius entry. For example, 30 mm is converted to 3 cm when you switch units. Vertical height value is the perpendicular apex-to-base distance, not the sloping edge. A value such as 4 is valid and, with a 3-unit radius, creates a 3 – 4 – 5 right triangle. Increasing height raises volume linearly and usually increases lateral and total area. Vertical height unit applies only to that field and can differ from the radius unit.

Slant height value is the straight distance from the apex to the rim of the base. It is optional when radius and vertical height are known. When used with radius, it must be greater than the radius; when used with vertical height, it must be greater than the vertical height. A common mistake is entering a slant height that cannot be the hypotenuse of the cone's right-triangle cross-section. Slant height unit converts that measurement independently. Result unit controls all displayed and exported results: lengths use the selected unit, areas use its square, and volume uses its cube. NIST's explanation of dimensional unit conversion explains why area conversion factors are squared and volume conversion factors are cubed.

Output guide

Total surface area (A) is the exact geometric identity πr² + πrl, displayed as an estimate rounded to four decimals. It includes both the base and curved side. Volume (V) is ⅓πr²h and measures enclosed three-dimensional space. A zero result is not shown because every accepted length must be positive. Lateral surface area (Aₗ) is πrl and excludes the base; this is often the relevant quantity for a paper cone or side covering. Base area (Aᵦ) is πr². The solved length card reports whichever of radius, vertical height, or slant height was left blank. The three summary pills report how many measurements were entered, which measurement was solved, and the active output units. In the detail table, Property names the quantity, Relationship shows the formula or whether the value was entered, and Current value gives the converted result.

Worked example

With the startup values r = 3 cm and h = 4 cm, the slant height is l = √(3² + 4²) = 5 cm. The base area is π × 3² = 28.2743 cm², the lateral area is π × 3 × 5 = 47.1239 cm², and the total surface area is 75.3982 cm². The volume is ⅓ × π × 3² × 4 = 37.6991 cm³. These values match the first-open cards, the detail table, and the downloadable workbook. OpenStax lists the same cone formulas for volume and surface area.