Centre square
What does Pythagoras' Theorem tell you about the radius of these circles?
Age
14 to 16
Challenge level
Problem
The diagram shows two identical large circles and two identical smaller circles whose centres are at the corners of a square.
The two large circles are touching, and they each touch the two smaller circles.
The radius of the small circles is $1cm$.
What is the radius of the large circles in centimetres?
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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Let $r$ be the radius of each of the larger circles.
The sides of the square are equal to $r+1$, the sum of the two radii.
The diagonal of the square is $2r$.
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By Pythagoras, $$(r+1)^2+(r+1)^2 = (2r)^2$$Simplifying gives: $$ 2(r+1)^2 = 4 r^2$$ i.e. $$(r+1)^2 = 2r^2$$
so$$r+1 = \sqrt{2}r$$
[$-\sqrt{2}r$ is not possible since $r+1> 0$].
Therefore $(\sqrt{2} - 1)r = 1$.
Hence $r= \frac {1}{\sqrt{2}-1} = \sqrt{2} + 1$.