Understand the right circular cone

A right circular cone has a circular base and an apex directly above its centre. Its vertical height h is perpendicular to the base, while slant height l follows the surface from apex to the base rim.

Properties that make the calculations work

The radius, vertical height and slant height form a right triangle, so l² = r²+h². The total surface area includes the circular base and the curved lateral area. Volume is one-third of the matching cylinder’s volume.

Where the model is useful

Select radius with height or slant height depending on your drawing. Inverse tools recover height from volume or total surface area. The model excludes oblique cones and truncated cones.

Measure the right thing

Use vertical height in the volume formula. Slant height must exceed the radius for a non-degenerate right cone.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k² and any volume by k³. This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area and cm³ for volume; the unit selector relabels rather than converts entered numbers.

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Wolfram MathWorld: Right circular cone provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.