Surface Area of a Cone Formula

Geometry Formulas: Area, Volume, Perimeter
Formula:

What is the formula for the surface area of a cone?

Answer

Surface area of a cone
S = π · r² + π · r · l

Here r is the radius, l is the slant height of a face, or the length along the solid. The result comes out in square units.

Where the formula comes from

S = π · r² + π · r · l

Cut the cone up one side and flatten it: the curved part opens into a sector of a circle of radius l, and its area works out to πrl. Add the base disc πr². If you know the height instead of the slant, l = √(r² + h²). Both steps assume a RIGHT circular cone — on a leaning one the slant is a different length on every side and neither formula holds.

Worked examples

Let r = 3, l = 5. Then the surface area of a cone ≈ 75.4 square units.

Let r = 2.5, l = 6.5. Then the surface area of a cone ≈ 70.69 square units.

The common mistake

Putting the height h in place of the slant l. Of the two, h is always the shorter — they are linked by l² = r² + h². The pyramid's surface area asks for a slant too, but a different one: the slant height of a triangular FACE, not of the whole solid.

Other quantities: Cone

Related Calculations

See Also

Geometry Formulas: Area, Volume, Perimeter

These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the surface area of a cone? (The answer is: S = π · r² + π · r · l). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.
Geometry Formulas: Area, Volume, Perimeter
Formula:

Values from the Formula (l = 5)

If r =Surface area
1≈ 18.85 square units
2≈ 43.98 square units
3≈ 75.4 square units
3.5≈ 93.46 square units
4≈ 113.1 square units
4.5≈ 134.3 square units

FAQ

What is the formula for the surface area of a cone?

Surface area of a cone S = π · r² + π · r · l

Where does this formula come from?

Cut the cone up one side and flatten it: the curved part opens into a sector of a circle of radius l, and its area works out to πrl. Add the base disc πr². If you know the height instead of the slant, l = √(r² + h²). Both steps assume a RIGHT circular cone — on a leaning one the slant is a different length on every side and neither formula holds.

What is the most common mistake here?

Putting the height h in place of the slant l. Of the two, h is always the shorter — they are linked by l² = r² + h². The pyramid's surface area asks for a slant too, but a different one: the slant height of a triangular FACE, not of the whole solid.

How do I use this formula?

Put your own numbers in place of the letters. With r = 2.5, l = 6.5, for instance, the formula S = π · r² + π · r · l gives ≈ 70.69 square units.

See Also