Formula and Method for the Radius of a Cone
A right circular cone is defined by three linked measurements: the base radius r, the perpendicular height h (from apex straight down to the base center), and the slant height l (the distance along the cone's surface from apex to the edge of the base). Because r, h, and l form a right triangle inside the cone, they always satisfy the Pythagorean relation l² = r² + h². This calculator uses that relation, together with the standard volume and surface-area formulas, to solve for the radius from whichever pair of values you already know.
How the calculation works
Pick which two quantities you know from the "Known Values" dropdown, then this calculator rearranges the matching formula to isolate r:
- Volume & height: starting from V = (1/3)πr²h, solve for r to get r = √(3V / πh).
- Slant height & height: from l² = r² + h², solve for r to get r = √(l² − h²).
- Lateral surface area & slant height: starting from L = πrl (the area of the curved side only), solve for r to get r = L / (πl).
- Total surface area & slant height: starting from S = πr² + πrl (curved side plus the circular base), rearrange into the quadratic πr² + πlr − S = 0 and solve with the quadratic formula, keeping the positive root: r = (−πl + √((πl)² + 4πS)) / (2π).
Once r is known, the calculator also recovers the height and slant height it wasn't given (again via l² = r² + h²) so it can report the cone's diameter, volume, and total surface area alongside the radius.
Common mistakes
- Height vs. slant height: the height h is the straight vertical distance to the apex; the slant height l runs along the cone's slanted surface and is always longer than h. Swapping them gives a wrong radius.
- Forgetting the 1/3 in the volume formula: a cone holds exactly one-third the volume of a cylinder with the same base and height — using V = πr²h instead of V = (1/3)πr²h triples the apparent radius.
- Lateral vs. total surface area: lateral surface area (L = πrl) covers only the curved side; total surface area (S = πr² + πrl) also includes the flat circular base. Using the wrong one gives the wrong radius.
- Impossible geometry: the slant height must always be greater than the height (l > h), since l is the hypotenuse of the right triangle formed by r and h. If you enter a slant height that isn't larger than the height, no real cone matches those numbers.
- Mixed units: enter volume in cubic units and area in square units of the same length unit you select — mixing feet and inches produces an incorrect radius.
Real-world applications
- Manufacturing traffic cones, funnels, and paper cups from flat sheet stock, where the lateral surface area determines the sector of material to cut.
- Sizing silos, hoppers, and conical tank bottoms from a target volume and available height.
- Architecture and tent design, where the slant height and height define the roofline and the radius sets the footprint.
- Packaging and mold design, where total surface area drives material and coating cost estimates.