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Geometry

Surface area of a cone

The circular base plus the curved surface, using the slant height l.

S=πr2+πrl
Curved surface onlyScurved=πrl
Slant heightl=r2+h2

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

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h = 4rl = 5
  • Slant height l = √(r² + h²) = 5
  • Volume = ⅓πr²h = 37.699 (a third of the cylinder around it)
  • Surface area = πr² + πrl = 75.398

What the symbols mean

S
the total surface area
r
the radius of the base
l
the slant height: from the edge of the base to the tip, along the surface

When to use it

Use it for the card in a party hat (curved part only), the material in a conical roof or tent, or the surface of a funnel. If you know the height instead of the slant height, get l from Pythagoras first.

Worked example

A cone has a base radius of 3 cm and a slant height of 5 cm. What is its total surface area? (r = 3, l = 5)

  1. Start from the formula.

    S=π⁢r2+π⁢r⁢l

  2. Substitute r = 3, l = 5 and work it out.

    S=π·32+π·3·5
    =π·9+π·3·5
    ≈75.3982

Answer S ≈ 75.40 cm²

Why it works

Cut the curved surface along a slant line and flatten it: it is a sector of a circle of radius l whose arc is the base circumference 2πr. A sector's area is half the radius times the arc: 12×l×2πr=πrl.

Common mistakes

  • Using the perpendicular height h in πrl. It needs the slant height, which is longer.
  • Adding the base for an open shape such as a hat or a funnel.