# Thread: Circle GCSE Challenge - Help needed

1. ## Circle GCSE Challenge - Help needed

Hello guys,

I put this in the wrong forum previously, so I apologise if you have seen this post before.

I've been trying to figure where to start with this, but no luck. I need help with this, because I've been at it for a while. I hope you understand what I'm getting at.

Here it is:

Q) The large circle has a radius of 10cm. Within the large circle are 4 smaller circles as shown in the link below (the image on the far left). 'You may have to click on the image for the picture to be clear so you can see what I mean.' Find the radius of the largest circle which will fit in the middle.

http://mathworld.wolfram.com/images/...rcles_1000.gif

2. Originally Posted by mathstudent
Hello guys,

I put this in the wrong forum previously, so I apologise if you have seen this post before.

I've been trying to figure where to start with this, but no luck. I need help with this, because I've been at it for a while. I hope you understand what I'm getting at.

Here it is:

Q) The large circle has a radius of 10cm. Within the large circle are 4 smaller circles as shown in the link below (the image on the far left). 'You may have to click on the image for the picture to be clear so you can see what I mean.' Find the radius of the largest circle which will fit in the middle.
1. I'll show you how to calculate the radius of the inner circle using the drawing with 5 smaller circles.

2. The tangent points of the smaller circles with the outer circle produce a regular pentagon. Let R denot the radius of the outer circle then the side of the pentagon is calculated as:

$\displaystyle s = 2 \cdot R \cdot \sin\left(\dfrac{180^\circ}{\underbrace{5}_{number \ of\ circles}}\right)$

3. Use proportion in the indicated isosceles triangle:

$\displaystyle \dfrac sR=\dfrac {2r}{R-r} \implies \boxed{r=\dfrac{sR}{2R+s}}$

4. The radius of the innermost circle, touching the smaller circle, is

$\displaystyle \rho=R-2r$

5. This method can be used with all other examples where more than 2 circles are surrounded by one big circle. You only have to change the number of circles in the equation at 2.

3. Join the centres of the small circles to form a square and extend one of its diagonals to form a diameter of the large circle. Call the radius of a small circle $\displaystyle R$ and calculate the length of the diagonal and hence the diameter of the large circle in terms of $\displaystyle R$. Put this equal to 20cm.

4. Thank you so much! Life saver!