this puzzle by Peter Grabarchuk appears on the 'Mathsisfun' website as 'double hearts ratio'

a square with semicircles

which area is larger

the purple

the gold yellow

this could be posed with a grid as a simple 'area of a circle' question

inviting students to make an initial estimate of which looks larger

however it is more challenging to pose the problem without a grid

various relationships are not too difficult to establish

what is the relationship between

R and r?

then the areas (in the right angled triangle version) are:

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