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Surface Area and Volume of a Cone

A cone's total surface area is the base circle (pi r squared) plus the unrolled lateral surface (pi r l); its volume is one-third pi r squared h, the same fraction that relates a cone to a matching cylinder. Learn both formulas, how slant height relates to radius and height, and a worked example.

What a cone is

A cone is a 3-D shape with a single circular base that narrows smoothly up to one point, called the apex. Every cone has three key measurements: the radius (r) of its circular base, its height (h, the straight up-down distance from base to apex), and its slant height (l, the distance along the outside surface from the edge of the base to the apex). These three are connected by the Pythagorean theorem: r² + h² = l².

The surface area formula

A cone's total surface area is made of two parts: the flat circular base, and the curved lateral (side) surface. Imagine unrolling the cone flat — the base stays a circle, and the lateral surface flattens into a sector (a pie-slice shape) with radius l.

A cone with radius 3, height 4, and slant height 5, and its net: a base circle (area πr²) plus a lateral sector (area πrl).

Base area: πr², the area of the circular base — see circumference of a circle for the circle basics behind it.
Lateral area: πrl, the area of the unrolled sector.
Total surface area: SA = πr² + πrl.

The volume formula

A cone's volume is exactly one-third the volume of a cylinder with the same base and height: V = (1/3)πr²h. That one-third relationship also appears in the volume of a cylinder, which a cone's volume is always compared against.

Worked example

Take a cone with radius 3 and height 4. First find the slant height with the Pythagorean theorem: l² = 3² + 4² = 9 + 16 = 25, so l = 5.

Surface area: SA = π(3)² + π(3)(5) = 9π + 15π = 24π ≈ 75.4 square units.
Volume: V = (1/3)π(3)²(4) = (1/3)π(36) = 12π ≈ 37.7 cubic units.

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Geometry