I’ll preface this with an apology if the question is too basic, too stupid or too weird.
I was reading Euclid’s definition of a circle - that is, “a circle is a plane figure bounded by one line, called the circumference, and is such that all straight lines drawn from a certain point inside the figure to the circumference are equal to each other.”
I wondered what would happen if the circumference of a circle with radius $r$ were stretched to form a straight line with a distance to the center being $r$, so that if you were to draw a dotted line from the center to either end of the newly formed line, it formed a triangle. My question is: would the length of the side be?
The length of one side would be $2\pi r$, and the height, as it were, would be $r$. How would you go about finding the side?
(Sorry again - this question is keeping me up).
To start welcome to SE, I started a few weeks ago so your not alone.
To visualize this, draw a diagram. If the line connecting the center to the line is perpendicular to the line (I assume that's what you are saying) then you have yourself a perpendicular bisector (the line splits it in two $r\pi$ on each side)! To find the length of the side you would use the Pythagorean Theorem ($a^2+b^2=c^2$)Plugging in the values, the side length comes out to be $r^2(1+\pi)$.
Hopes that helps!!