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Giant Holly Leaf

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).


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.

Get Cross

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?

Two Shapes & Printer Ink

Age 14 to 16
Challenge Level
Some solution we were offered cleverly reasoned that the grid, offered in the Hint as an approximation, was in fact completely accurate at the pixel level - which for printer ink is what counts. Sometimes we explore mathematics in unreal perfection for the sake of the thinking that will draw us into. Other solutions did this:

"You can split the problem up into combinations of circular shapes, each of which we can find the area of. Doing this I foundthat the star is over 5% larger than the S shape."