Our WordPress site was a little out of date, but we did not want to update it until we had some free time. We have a bit of time now, so we downloaded copies of the website as a backup and tried updating to the latest version of WordPress.

If you tried to get on the site last night, you might have noticed that it was not there. The upgade did not go well, so it is a good thing we have backups. Opinions vary on whether WordPress version 7.1 is working, but the plugins are certainly not updated. Also, there appears to be a bug that makes all plugins report they are broken.

Since it broke some of the site functionality and appearance, we used the backups to downgrade and updated to WordPress 7.0.4, which was quite a pain in the rear. Apparently, you are only supposed to update to the latest version and just suck it up until they fix everything.

We are currently at version 7.0.4, which appears to be the last stable version with working plug-ins. If you notice any issues, post a comment about it.

The best part is that we can now…

Lets see if we can calculate the area of a circle! When calculating areas, you have to integrate across infinitesimal areas. Normally, you would have an area calculated as:

An integral to calculate area is the summation of tiny areas taken to the infinite limit. The tiny areas vary based on the coordinate system you select, and the image below shows rectangular and polar coordinates. The red is the infinitesimal area being summed.

For a rectangle, you integrate from zero to the length and zero to the width using a fixed size area element, dx · dy. In calculus, angles are measured in radians, zero to 2π, instead of degrees, zero to 360. For a circle, you need to integrate over the radius from zero to R and over the angles from zero to 2π using an incremental area that gets larger the further you are from the center, r · dr · dθ. The reason for this is that the wedge gets wider the further you get from the center of the circle.

In calculus, this is written as:

In general, the integral of a function is memorized or looked up in a chart. Some integrals have to be estimated using numerical methods instead of functions. Polynomials are straight forward. Solving the integrals gives the area of a circle.

Now you know how the area of a circle is derived!

As a bonus, the value of Pi (π) is not apple, blueberry, or lemon meringue, it is the ratio of the circumference of a circle to the diameter of a circle. It is the same for all circles. You were probably losing sleep wondering about that.

Note: Math text has been changed to red to address an issue where mobile browsers using force dark mode make the math unreadable. In the future, we should implement a light/dark mode that may address the issue.