Radius of a Cone Calculator
Find a right circular cone's radius from common dimension pairs, then calculate its complete geometry when enough information is available.
Known cone dimensions
Enter a radius or base area, or a supported pair of independent measurements. Leave unknown fields blank.
Changing units converts every entered value and preserves the physical cone.
Perpendicular distance from the base center to the apex.
Distance along the cone's side from the rim to the apex.
Enter directly when known; it is also the calculator's primary result.
Total area of the circular base plus the curved side.
Three-dimensional space inside the cone.
Area of the curved side only, excluding the base.
Area of the circular base; this alone determines the radius.
Calculated geometry
Results update as you type. Blank secondary results mean another independent dimension is needed.
A complete right circular cone is determined by the entered height and slant height.
Dimension details
Every value below comes from the same canonical cone model used by the on-page results and Excel workbook.
| Quantity | Symbol | Calculated value | Relationship |
|---|---|---|---|
| Radius | r | 5.0000 cm | √(l² – h²) |
| Height | h | 12.0000 cm | Entered value |
| Slant height | l | 13.0000 cm | Entered value |
| Total surface area | A | 282.7433 cm² | πr(r + l) |
| Volume | V | 314.1593 cm³ | πr²h / 3 |
| Lateral surface area | A_L | 204.2035 cm² | πrl |
| Base area | A_B | 78.5398 cm² | πr² |
Length, area, and volume use the selected unit raised to the first, second, and third power. Results are displayed to four decimal places while calculations retain full precision.
How to use the radius of a cone calculator
What this calculator does
This calculator estimates the radius of a nondegenerate right circular cone from measurements that mathematically determine it. It can also reconstruct height, slant height, total surface area, volume, lateral surface area, and base area when the entered information defines a complete cone. It does not identify the radius of an oblique cone, a truncated cone, or an object whose cross-section is not circular. The model assumes exact geometric measurements and does not include manufacturing tolerances, wall thickness, rounding allowances, or material deformation.
When to use it
Use it when checking a geometry exercise based on height and slant height, converting a known base area into radius, deriving radius from volume and height, or validating dimensions for a conical part, vessel, funnel, pile, or pattern. It is also useful for cross-checking whether several measured values could belong to the same right circular cone.
How to calculate
- The calculator opens with a ready-to-use 12 cm height and 13 cm slant height demonstration. Its complete results and a validated example Excel workbook are immediately available.
- Choose the Measurement unit. Length fields use that unit, area fields use its square, and volume uses its cube. Changing the unit converts existing entries rather than relabeling them.
- Replace the sample values with measurements you know. Enter Radius (r) directly, enter Base area (AB) by itself, or provide a supported pair such as height plus slant height, height plus volume, slant height plus lateral area, or slant height plus total surface area.
- Read Radius (r) as the primary result. The summary pills identify the solution basis and whether a complete geometry was found. Review the secondary cards and Dimension details table for the remaining measurements.
- Select Download Excel to export the current validated model. Select Reset to clear the demonstration and all entries. Reset may disable the download until a complete valid radius solution is entered again.
The formulas use standard right-cone relationships. OpenStax derives the cone volume identity V = πr²h/3 in its volume-by-slicing explanation. For unit interpretation, NIST explains that length is one-dimensional, area is squared, and volume is cubed in its dimensional unit-conversion guidance.
Input guide
Measurement unit is required and accepts millimeters, centimeters, meters, inches, or feet. The sample uses centimeters. A unit change affects every entered number according to its dimension; a common mistake is manually converting only a length while leaving an area or volume unchanged.
Height (h) is an optional positive decimal length, such as 12 cm. It is perpendicular to the base, not measured along the side. Increasing height while keeping slant height fixed reduces the radius. Slant height (l) is an optional positive decimal length, such as 13 cm. For a real cone solved from height and slant height, slant height must exceed height.
Radius (r) is an optional positive decimal length. Entering 5 cm immediately establishes the primary result; one additional independent dimension is needed to determine the full cone. Do not enter diameter in this field. Surface area (A) is an optional positive decimal area that includes both the curved side and circular base. With slant height, it solves the quadratic relationship A = πr(r + l).
Volume (V) is an optional positive decimal cubic measure. With height, it gives r = √(3V/(πh)). A frequent mistake is supplying liters while the selected unit is centimeters; enter cubic centimeters instead. Lateral surface area (AL) is the curved side only and excludes the base. With slant height, it gives r = AL/(πl).
Base area (AB) is an optional positive decimal area of the circular base. It alone determines radius through r = √(AB/π). NIST identifies the square meter as the SI derived unit for area in its SI area guidance. The calculator rejects commas, scientific notation, zero, negative values, and unsupported symbols so ambiguous input is not silently reinterpreted.
Output guide
Radius (r) is the exact geometric radius implied by the accepted measurements, displayed in the selected length unit. The Radius, solution basis, and status pills summarize the same model. “Complete geometry” means height, radius, and slant height are all known; “Radius available” means the radius is valid but another independent dimension is needed.
Height (h) and Slant height (l) are linear dimensions. Surface area (A), Lateral area (AL), and Base area (AB) are square-unit outputs. Volume (V) is a cubic-unit output. A missing secondary value is not zero; it means the current inputs do not uniquely determine that quantity. The Formula used field shows the relationship that established the radius, and the Dimension details table lists each quantity, symbol, calculated value, and governing relationship.
Worked example
With height h = 12 cm and slant height l = 13 cm, the radius follows the Pythagorean relationship: r = √(13² – 12²) = √(169 – 144) = √25 = 5 cm. The base area is π × 5² = 78.5398 cm². The lateral area is π × 5 × 13 = 204.2035 cm². Total surface area is their sum, 282.7433 cm², and volume is π × 5² × 12 / 3 = 314.1593 cm³. These values match the calculator's first-open display and workbook checkpoints.
Formula selection and consistency checks
The calculator selects a formula only when the entered values form a supported, non-ambiguous relationship. If you enter more measurements than necessary, it verifies them against the reconstructed cone. A mismatch produces an error instead of averaging values or silently choosing one. This is important because real measurements may be rounded or may describe an oblique or imperfect object rather than an ideal right circular cone.