So, The other day I was playing around with Geogabra, as one does, when I made a discovery I couldn't find elsewhere. I found that if you have 45-45-90 special right triangles within a circle, you can do some math to find the area of the circle if your only given the area of the triangle, and vice-versa. This is already somewhat easy if you know basic algebra, but there is a way to do it by just multipling and dividing one time. Let me explain.
If you have an image like this, where the triangle within it is a 45-45-90 triangle, and the congrent sides are equivelent to the radius of the circle. (The areas are already filled in for easyness sake): Image for Example.
You can find the area of the circle if you're given the area of the triangle by multipling the area of the triangle by 2(pi).
The example given in the image, the area of the triangle is 50, so 50*pi = about 314.16, which is about the area of the circle. Dividing the area of the cricle by 2(pi) would give you 50, the area of the triangle.
You can do something similar with a 45-45-90 triangle, but the hypotenuse is equal to the diameter of the circle: Image 2 for Example
to solve this, simple multiply the area of the triangle, in this case 100, by pi to get the circle's area, or divide the area of the circle by pi to get the triangle's area.
This works for any triangle/circle combonation that follow these rules:
- The triangle is 45-45-90
- Either the hypotenuse is equal to the circle's diameter, or the other two sides are both equal to the circle's radius.
I hope you enjoyed this math jargin. My question is: is this an actal thing that already exists and people know about, or is it something that I discovered? I dought that, but hey, it's worth a shot. I'm asking because I'm currently taking geometry, and we are long past our special right triangle unit, but I have never heard of this before. Thanks for coming to my Ted Talk.