Calculate the radius of a cone from volume & height, surface area & slant height, lateral area, base area, or circumference. Shows all cone properties with visual breakdowns.
The radius of a cone is the distance from the center of its circular base to any point on the base's edge. It is one of the key dimensions needed to calculate a cone's volume, surface area, slant height, and other properties. Whether you are working on an engineering design, making a party hat, or solving a geometry problem, knowing how to find the cone's radius from other measurements is essential.
This calculator supports five different solving methods. You can determine the radius using the volume and height, the total surface area and slant height, the lateral (curved) surface area and slant height, the base area alone, or the base circumference alone. Each method applies a different rearrangement of the standard cone formulas so you always have a path to the answer with whatever data you have.
A right circular cone is defined by two independent measurements — usually the base radius and the height. From these, every other property follows. The slant height is √(r² + h²), the volume is (1/3)πr²h, the lateral surface area is πrl, and the total surface area adds the base area πr² on top. All of these relationships are reversible, which is what this tool exploits.
Use the presets to explore real-world cone shapes like traffic cones, ice cream cones, party hats, and funnels. The surface breakdown bars give you an instant visual sense of how the base area compares to the lateral area, and the reference table lists common cone dimensions for quick comparison.
Use this calculator when you know a cone's volume, slant height, lateral area, or base measurements but need the missing radius quickly. It is useful for geometry homework, sheet-metal pattern work, packaging design, funnel sizing, and any situation where you need to move from a partial cone specification to a complete set of dimensions without manually rearranging several formulas.
r = √(3V/(πh)) | r = (−πl + √(π²l² + 4πSA))/(2π) | r = LA/(πl) | r = √(A/π) | r = C/(2π), where V = volume, h = height, SA = surface area, l = slant height, LA = lateral area, A = base area, C = base circumference.
Result: Radius = 9.7721 cm
Given a cone volume of 1000 cm^3 and height of 10 cm, the calculator uses r = sqrt(3V / (pi h)). Substituting the inputs gives r = sqrt(3000 / (pi x 10)) = sqrt(95.493) = 9.7721 cm. From that radius, the cone has a slant height of about 13.9739 cm and a total surface area of about 732.6387 cm^2.
A right circular cone is defined by three measurements: the **base radius r**, the **height h** (perpendicular distance from base to apex), and the **slant height l** = √(r² + h²) (distance along the surface from apex to any base-edge point). Each of the cone's area and volume formulas involves the radius: **volume** V = πr²h/3 (one-third of the enclosing cylinder), **lateral surface area** A_lat = πrl, **total surface area** A_tot = πrl + πr² = πr(l + r).
This calculator inverts these formulas to find r from whatever you know. Given V and h: r = √(3V / (πh)). Given total surface area and slant height: the equation πr(l + r) = A_tot is quadratic in r, solved as r = [−πl + √(π²l² + 4πA_tot)] / (2π). Given lateral area and slant height: r = A_lat / (πl). Given base circumference C: r = C / (2π). Given base area A_base: r = √(A_base / π).
When a cone is cut by a plane parallel to the base, it produces a **frustum** with two circular faces of radii r₁ and r₂. The slant height of the frustum is l = √((r₁ − r₂)² + h²), the lateral area is π(r₁ + r₂)l, and the volume is πh(r₁² + r₁r₂ + r₂²)/3. Frustums appear in buckets, drinking cups, lampshades, and cooling towers.
Cone radius calculations are central to **manufacturing** (CNC machining of tapered bores, die design), **civil engineering** (conical water tanks, hoppers, silos), **geology** (estimating the volume of a volcanic cone from topographic maps), and **food science** (ice-cream cone capacity). In **optics**, the cone half-angle determines the numerical aperture of a lens, so understanding cone geometry is also relevant to microscopy and telescope design.
The radius of a cone is the distance from the center of the circular base to any point on the base edge. It determines the width of the cone at its base.
Use the formula r = √(3V / (πh)). Multiply the volume by 3, divide by π times the height, then take the square root.
The slant height is the distance from the tip (apex) of the cone to any point on the edge of the base, measured along the surface. It equals √(r² + h²).
Not from the lateral area alone — you also need the slant height. The formula is r = LA / (πl).
The lateral area is just the curved surface of the cone (πrl). The total surface area adds the flat circular base (πr²): SA = πrl + πr².
This calculator assumes a right circular cone (apex directly above the center of the base). For oblique cones, the formulas are more complex and depend on the tilt angle.