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# Six Circles

The small rectangle consists of $12$ of the radii of the circles, each connecting a point of contact to the centre of the relevant circle. These are shown in green and orange in the diagram on the right.

Since this has a total length of $60\text{cm}$, each radius is of length $60\text{cm} \div 12 = 5\text{cm}$.

The large rectangle can also be broken down into segments of this length. These are shown in blue and red on the diagram. There are $20$ of these, so the perimeter of the large rectangle is $5\text{cm} \times 20 = 100\text{cm} = 1\text{m}$.

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Age 11 to 14

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The small rectangle consists of $12$ of the radii of the circles, each connecting a point of contact to the centre of the relevant circle. These are shown in green and orange in the diagram on the right.

Since this has a total length of $60\text{cm}$, each radius is of length $60\text{cm} \div 12 = 5\text{cm}$.

The large rectangle can also be broken down into segments of this length. These are shown in blue and red on the diagram. There are $20$ of these, so the perimeter of the large rectangle is $5\text{cm} \times 20 = 100\text{cm} = 1\text{m}$.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?