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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

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A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Curvy Areas

Age 14 to 16
Challenge Level

Take a look at the image below:

Can you see how the image was created?
Try to recreate it using a ruler and compasses.

Here are two images created in a similar way.

Can you work out the proportion of the 3-colour, 4-colour and 5-colour circles which is shaded red?
Can you make any generalisations?
Can you prove your ideas?

What about the proportion which is shaded orange? Yellow? ... 
Can you make any generalisations?
Can you prove your ideas? 

Click here for a poster of this problem.