The total surface area of a cone consists of two parts:

  • The curved surface area

  • The flat surface area (of the base)

The flat surface area is simply \(\pi {r^2}\), since the base is circular. What is the curved surface area? To determine that, visualize what would happen if we cut open the cone and roll it out:

3D and planar shapes of cone

We will obtain a circular sector. It would be a good idea to try this activity. Cut out a circular sector from a piece of paper. Then, fold / roll the paper so that the two straight edges align with each other, and you will get a cone.

Note that the radius R of the circular sector is the slant height of the cone. Denote the slant height by  \(l\). Observe that \(l= \sqrt {{r^2} + {h^2}} \). Also, the arc length L of the circular sector is the circumference of the cone’s base, which is \(2\pi r\). Now, the area of the circular sector is

\[\begin{align}&A = \frac{1}{{2\pi }} \times \frac{L}{R} \times \pi {R^2}\,\,\,\left( {{\rm{why}}?} \right)\\&\,\,\,\,\, = \frac{{LR}}{2} = \frac{{\left( {2\pi r} \right)\left( l \right)}}{2}\\&\,\,\,\,\, = \pi rl\end{align}\]

This is also the curved surface area of the cone!

To summarize:

  • Flat SA= \(\pi {r^2}\)

  • Curved SA = \(\pi rl = \pi r\sqrt {{r^2} + {h^2}} \)

  • Total SA = \(\pi {r^2} + \pi rl = \pi r\left( {r + l} \right)\)

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