Combine a cone's circular base with its slanted lateral surface. A checkable formula accompanies total surface area instead of leaving an unexplained number.
A right cone's lateral area depends on slant height l, not perpendicular height h. The Pythagorean relation l = √(r² + h²) bridges the two measurements, and total area is πr² + πrl.
This definition sets the boundary between cone surface area and a neighboring calculation with similar inputs.
Square radius and height, add, and take the square root for slant height. Multiply πrl for the curved surface, then add πr² for the base. A hand-worked extension of Cone Surface Area is slant-height relation.
Radius 5 and height 12 form a 5–12–13 right triangle, so slant height is 13. Total area is 25π + 65π = 90π, about 282.7433 square units.
Cone surface estimates arise in funnels, shades, hoppers, party hats, and tapered covers. Whether the circular base exists changes the required material. A related application of Cone Surface Area is cone volume.
Do not substitute vertical height directly for slant height in lateral area. This calculator includes the base; an open cone uses only πrl. If the Cone Surface Area assumptions do not fit, consider cylinder surface.
Changing only one field helps distinguish a data-entry problem from the intended behavior of cone surface area.
A field name is part of the formula. Match the problem's quantities to radius and perpendicular height, then check that they share the scale assumed by the cone surface area relationship.
Height first changes slant height through a square root, then lateral area through that slant. Radius influences the base, the slant, and the lateral multiplier at once. This is a stronger check than judging the answer only by how many decimal places it shows.
Match the reported precision to radius and perpendicular height, not to the number of digits the browser can display. Preserve an exact form when it communicates the cone surface area structure more clearly than a decimal.
Perpendicular height belongs in volume, while slant height belongs in lateral surface. This page derives the latter before evaluating the covering area. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: radius and perpendicular height, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
It runs diagonally from the rim to the apex, forming the hypotenuse of a right triangle.
Yes. Subtract πr² if the physical cone is open.
The lateral formula can be applied directly, though this page asks for perpendicular height.
Yes. Convert them to one linear unit before calculating.