Right Circular Cone Calculator

A right circular cone has a circular base and an apex positioned directly above the center of that base. The radius r measures from the center of the base to its edge, while the perpendicular height h runs from the base center to the apex. These two measurements completely determine the cone's slant height, base area, lateral area, total surface area, and volume. The slant height is the straight-line distance from the apex to any point on the circular rim. A cross-section through the cone's axis creates a right triangle with legs r and h, so the Pythagorean theorem gives slant height s = √(r² + h²). This distinction matters: the perpendicular height is used for volume, but the slant height is used for the curved lateral surface. The circular base area is πr². The lateral area, representing only the curved surface, is πrs. Adding the base gives total surface area πr² + πrs, often written πr(r + s). Volume is one third of the corresponding cylinder's volume, so V = πr²h / 3. The calculator retains π throughout the calculation and rounds only the displayed values to six decimal places. Keep radius and height in the same unit. If both are entered in centimeters, slant height is in centimeters, areas are in square centimeters, and volume is in cubic centimeters. Mixing inches and feet without conversion produces a result with no coherent unit. A diameter must also be divided by two before it is entered as the radius. Cone calculations appear in geometry exercises and in practical estimates for funnels, conical roofs, piles of material, paper cups, traffic cones, and manufactured parts. Volume estimates capacity or material quantity, while lateral area estimates material needed for the curved shell. Total surface area includes one circular base; an open cone may require only lateral area. Real objects may have wall thickness, a truncated top, or seams, so engineering and fabrication work should add the appropriate allowances after finding the ideal geometric measurement. A useful check is scaling. If every linear dimension doubles, the slant height doubles, each area becomes four times as large, and volume becomes eight times as large. This relationship helps identify input mistakes and confirms that a calculated result has the expected order of magnitude.