Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362552 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0546 1213432 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0546 1213432 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1904 4134632 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1918 4162088 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1925 4179800 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1927 4210320 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1937 4233040 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1937 4233040 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2466 4902888 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2500 5095040 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2506 5115968 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2622 5319992 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2623 5320248 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Area of a Regular Polygon in a Circle of Radius r

Suppose a regular polygon with  
\[n\]
  sides is inscribed in a circle of radius  
\[r\]
, with the vertices of the polygon touching the circle.
From the centre of the circle lines are drawn to each vertex, and  
\[n\]
  triangles are formed. The angle of each triangle at the centre of the circle is  
\[\frac{360}{n}\]
, and the area of each triangle is  
\[\frac{1}{2}r^2 sin (\frac{360}{n})\]
.

There are  
\[n\]
  such triangles, so the area of the polygon is  
\[\frac{n}{2}r^2 sin (\frac{360}{n})\]
.
As the number of sides gets larger  
\[sin (\frac{360}{n}) \]
  tends to  
\[\frac{2 \pi}{n}\]
, so the area of the polygon tends to  
\[\frac{n}{2} r^2 \times \frac{2 \pi}{n}= \pi r^2\]
.