Surface Area of a Cone

Statement

The surface area of a cone is the sum of the curved (lateral) surface area and the base area:

\[ S = \pi r \ell + \pi r^2 \]

where \(r\) is the radius and \(\ell\) is the slant height of the cone.

Why it’s true

• The curved surface of a cone unfolds into a sector of a circle with radius \(\ell\). Its area is \(\pi r \ell\).
• The circular base has area \(\pi r^2\).
• Adding them gives the total surface area.

Recipe (how to use it)

1. Identify the radius \(r\) and slant height \(\ell\).
2. Calculate the curved surface area using \(\pi r \ell\).
3. Calculate the base area using \(\pi r^2\).
4. Add the two areas to get the total surface area.

Spotting it

Look for words like “surface area of a cone”, “total surface area”, or a problem giving radius and slant height.

Common pairings

• Volume of a cone \(\tfrac{1}{3}\pi r^2 h\).
• Pythagoras’ theorem if slant height is not given (\(\ell = \sqrt{r^2 + h^2}\)).

Mini examples

1. Given: \(r=3\), \(\ell=5\). Answer: \(S=\pi(3)(5)+\pi(3^2)=15\pi+9\pi=24\pi\).
2. Given: \(r=7\), \(\ell=25\). Answer: \(S=\pi(7)(25)+\pi(49)=175\pi+49\pi=224\pi\).

Pitfalls

• Confusing slant height with vertical height.
• Forgetting to include the base area.
• Mixing units (e.g. cm with m).

Exam strategy

• Check whether the question asks for curved surface area only or total surface area.
• Write down both terms clearly before adding.
• Always square the radius only when calculating \(\pi r^2\).

Summary

The surface area of a cone is found with \(S = \pi r \ell + \pi r^2\). The first term is the curved surface, the second is the base area.