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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."