Radius of a Cone Calculator

The radius of a cone is derived using standard geometric formulas linking the circular base to the three-dimensional body. Depending on your known inputs, the calculator applies specific relationships: for instance, when dealing with volume (V) and height (h), the formula is derived from V = (1/3) * π * r^2 * h, which rearranges to solve for radius as r = sqrt((3 * V) / (π * h)). If you know the base area (A), the radius is calculated directly from the circle area formula: r = sqrt(A / π). For lateral surface area (L) and slant height (s), the radius relies on L = π * r * s, simplifying to r = L / (π * s).

Imagine you need to find the radius of a cone that has a known volume of 150 cubic centimeters and a vertical height of 8 centimeters. First, substitute your values into the rearranged volume formula: r = sqrt((3 * 150) / (π * 8)). Next, multiply the numerator to get 450. Then, multiply the denominator: 8 * 3.14159 yields approximately 25.1327. Divide 450 by 25.1327 to get approximately 17.9048. Finally, take the square root of 17.9048, which gives you a final cone radius of roughly 4.23 centimeters.

Always double-check your units of measurement before entering numbers into the calculator; mixing centimeters and meters will result in massive calculation errors. Remember the critical distinction between slant height (the diagonal length along the side) and vertical height (the straight-line distance from the peak down to the exact center of the base). If you only have total surface area rather than lateral area, keep in mind that total surface area includes both the base circle and the curved side.