# Thread: Intersection of circle and square

1. ## Intersection of circle and square

Consider the system of equations
x^2 + y^2 = R^2 (circle)
|x| + |y| = 1 (square)
I need to find the number of solutions to the two equations for different values of R.
I know if R>1 there are 0 solutions, 4 if R=1. But there is a value, t, of R for which the circle intersects the square 8 times i.e. t<R<1. My problem is how to find the value t?

2. There is actually a range of values t for which the square and the circle will intersect in 8 locations. There is another value of t aside from 1 where the square and the circle intersect in 4 locations. Determine this value, and call it c. What happens for values of t such that c<t<1?

3. Hi

Yes, the other point is where c=t....I know the answer for t is sqrt2 / 2 but I can't figure out how to get that value...

4. Originally Posted by bigdoggy
Hi

Yes, the other point is where c=t....I know the answer for t is sqrt2 / 2 but I can't figure out how to get that value...
That is the value for which the circle is inscribed in the square. The radius of the circle is the distance from the center of the square to a midpoint of one of the sides of the square.

5. Look at the diagram. The circles have radii of 0.8, 1, & 1.2.
What is the distance from (0,0) to (1,1)?

6. Originally Posted by Plato
Look at the diagram. The circles have radii of 0.8, 1, & 1.2.
What is the distance from (0,0) to (1,1)?
The distance is sqrt2...

7. Originally Posted by icemanfan
The radius of the circle is the distance from the center of the square to a midpoint of one of the sides of the square.
Why is this?