Question

Get an approximation of pi using a square and triangles inside a circle.

Answer 1

The idea is to first calculate the area of the square and then add triangles.

Answer 2

Asquare = (2r)^2 = 2
My problem is with getting the first level of triangles, and then adding triangles to those triangles

Answer 3

what did you pick for r ? how about starting with r= 1?

Answer 4

I'm using r for distance from center to square. The radius is 1.

Answer 5

So with that I can get a height for a triangle 1-r.

Answer 6

ok, but using r for the length of 1/2 the side of the square is confusing (people see "r" and think radius)

can I use x instead of r?
diameter is 2, and this is the length of the diagonal of the square
so its side is 2/sqr(2) = sqr(2)
and x= 1/2 of a side = sqr(2)/2

Answer 7

Ok, lets use x instead.

Answer 8

the height of the triangle will be r- x
1 - \sqr(2)/2

Answer 9

Ok, and base would be x*2,right?

Answer 10

yes

Answer 11

Ok. So that would be ((1 - sqrt(2) / 2)(sqrt(2))/2. That times 4 to get total area

Answer 12

yes

Answer 13

Ok, I was good until here. I started to get lost when adding a new layer of triangles

Answer 14

So I guess it will pass through the half of the half of the base of the triangle, and then half of the side of the triangle

Answer 15

yes, but now it's not clear what the altitude of the new triangle is.
the base of the new triangle is the "hypotenuse" of the old triangle, so we can find it using pythagoras

Answer 16

Ok, give me a minute, I'll try to get it

Answer 17

Ok. The base is pretty simple. The altitude is a little harder.

First, I would calculate the distance between the center and the intersection with the square. Then add the intersection with the triangle.

For the square, I did:


So a = sqrt((x/2)^2 + x^2)