Surface area of a cone

The surface area of a cone is also straightforward to calculate once you understand the 2D shapes that make up the surface.

The bottom part is a circle and so the surface area of that part is $\pi r^2$

The curved part of the cone above the base is like this. The two red lines join together to make the cone shape and the blue arc curls round to become a circle with radius r.

This shape is a sector of a circle with radius l as shown below. The area of the circle is $\pi l^2$ . The length of the blue arc is the circumference of the blue circle which is $2\pi r$ . The circumference of the large circle is $2\pi l$ .

The proportion of the circle taken up by the sector is the same proportion that the arc of the sector is of the circumference of the circle. This is $\frac{2\pi r}{2 \pi l} = \frac{r}{l}$

The area of the sector is the proportion of the circle taken up by the sector multiplied by the area of the circle.

This is $\frac{r}{l} \times \pi l^2$

$= \pi rl$

The surface area of a cone is therefore $= \pi rl + \pi r^2$

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