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title: Surface Area of a Cone
---
## Surface Area of a Cone

A cone is a three-dimensional solid that has a circular base, which is connected by a curved surface to its vertex. The curved surface of the cone is formed by a set of line segments that connect the vertex to the circumference of the circle at the bottom. 

The `radius of the cone 'r'` is defined by the radius of the circle formed at the bottom. The `slant height 'l'` is the distance from any point on the circle to the vertex of the cone. Lastly, `the altitude 'h'` is defined by the distance measured from the vertex to the cirle's center.

* The `slant height 'h'` is calculated as:
<a href="http://www.codecogs.com/eqnedit.php?latex=l&space;=&space;\sqrt{r^2&plus;a^2}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?l&space;=&space;\sqrt{r^2&plus;a^2}" title="l = \sqrt{r^2+a^2}" /></a>

* The `lateral surface area` is given by: <a href="http://www.codecogs.com/eqnedit.php?latex=L&space;=&space;\pi&space;r&space;l&space;=&space;\pi&space;r&space;\sqrt{r^2&plus;h^2}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?L&space;=&space;\pi&space;r&space;l&space;=&space;\pi&space;r&space;\sqrt{r^2&plus;h^2}" title="L = \pi r l = \pi r \sqrt{r^2+h^2}" /></a>

* The `base surface area` is given by: <a href="http://www.codecogs.com/eqnedit.php?latex=B&space;=&space;\pi&space;r^2" target="_blank"><img src="http://latex.codecogs.com/gif.latex?B&space;=&space;\pi&space;r^2" title="B = \pi r^2" /></a>

Hence, the total surface area of the cone is the sum of the `lateral surface area` and the `base surface area`.

<a href="http://www.codecogs.com/eqnedit.php?latex=A=&space;L&space;&plus;&space;B&space;=&space;\pi&space;r&space;l&space;&plus;&space;\pi&space;r^2&space;=&space;\pi&space;r&space;(l&space;&plus;&space;\pi&space;r)&space;=&space;\pi&space;r&space;(\sqrt{r^2&plus;h^2}&plus;&space;\pi&space;r)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?A=&space;L&space;&plus;&space;B&space;=&space;\pi&space;r&space;l&space;&plus;&space;\pi&space;r^2&space;=&space;\pi&space;r&space;(l&space;&plus;&space;\pi&space;r)&space;=&space;\pi&space;r&space;(\sqrt{r^2&plus;h^2}&plus;&space;\pi&space;r)" title="A= L + B = \pi r l + \pi r^2 = \pi r (l + \pi r) = \pi r (\sqrt{r^2+h^2}+ \pi r)" /></a>

<img src="https://qph.fs.quoracdn.net/main-qimg-7e85a657d49edce7ee0b914024017acf.webp">

#### More Information:

* The derivation for the formula can be found at - [mathopenref.com](https://www.mathopenref.com/coneareaderivation.html)
* The link to a video explaining the derivation is given below:

<a href="http://www.youtube.com/watch?feature=player_embedded&v=rd8tbD2eekM
        " target="_blank"><img style="display: block; margin: 0 auto"  src="http://img.youtube.com/vi/rd8tbD2eekM/0.jpg" 
        alt="Video Thumbnail Image" width="400" border="2"/></a>
