Look a little closer.

Separate slant height from vertical height. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Unrolling the curved surface forms a circular sector of radius l.

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.

02 / A different perspective

The sector’s arc length equals the base circumference.

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.

03 / A connection to explore

A cone and cylinder with the same base and height have volumes in the ratio 1:3.

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.

Try the idea with numbers

Open the radius and vertical height calculator, note its outputs, then multiply each length input by two. Volume should increase by eight and surface area by four, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Right circular cone at Wolfram MathWorld.