Just a while ago, while feeling bored, I took a pen and pencil, and started doing some math. After a while of doing something I found out that the hypotenuse of half the triangle (formed from a circle) ≈ pi * r Here's how I got to that:
- Convert the circle into a triangle
- Now, the radius is the height of the triangle, and the circumference is the base of the triangle
- Next we replace the radius with r (for simplicity), and replace the circumference with its formula (2 pi * r)
- Next we divide the triangle in half (making identical 2 triangles)
- Now the base of the triangle is half (pi * r), but the height is the same (r)
- Now we try to calculate the hypotenuse --
h = sqrt( (r)^2 + (pi * r)^2 )Note: sqrt means square root
- Now we add in the values and calculate
For some reason, the hypotenuse ≈ pi * height
For example:
radius(h) = 3 cm
sqrt( (r)^2 + (pi * r)^2 )
= sqrt( (3)^2 + (3.14 * 3)^2 )
= sqrt( 9 + (9.42)^2 )
= sqrt( 9 + 88.7364 )
= sqrt( 97.7364 )
= 9.88617216116
Which approximately equals: pi * r = 3.14 * 3 = 9.42
Please tell me if I'm missing something super obvious.
You appear to be forming an isosceles triangle by "unrolling" a circle a distance equal to its circumference to form the base of a triangle with height equaling the radius.
Then the length of the two congruent sides equals $\sqrt{r^2+\pi^2r^2}=r\sqrt{1+\pi^2}$.
Can you see now why this "approximately" equals $\pi r$?