Archimedes Laboratory Project™
Puzzles & Mental Activities that Enhance Critical and Creative Thinking Skills

	The pink cat-claw shape is bounded by 3 arcs of circle with same diameter. The perimeter of the green parallelogram is 42 cm long and its height is half the radius of the circles. Then, the area of the pink shape is: larger, smaller or equal to the area of the green parallelogram? La forme rose ressemblant à une griffe de chat est délimitée par trois arcs de cercle ayant même diamètre. Le périmètre du parallélogramme vert est de 42 cm et sa hauteur correspond à la moitié du rayon des cercles. Donc, la zone de la forme curvilinéaire rose est-elle: supérieure, inférieure ou égale à l'aire du parallélogramme vert? La forma rosa ad artiglio di gatto è delimitata da tre archi di cerchio di diametro uguale. Il perimetro del parallelogramma verde è di 42 cm e la sua altezza, la metà del raggio dei cerchi. Quindi, l'area della forma curvilinea rosa è: maggiore, minore o uguale all'area del parallelogramma verde? You can comment and discuss this puzzle on our FaceBook web page. › Click to show/hide the solution Solution: There is actually no need to know the size of any shape to solve this puzzle! Subdivide the shapes into 7 respective regions (named s1 , s2 , s3 , s4 , s5 ), as shown in the diagram below. Area of the green parallelogram AO1O2B: 2s1 + s2 + s5 Area of the pink shape ABC: s4 - s3 + s5 By similitude: s4 = 2s1 + s2 + s3 Then: s4 - s3 + s5 = (2s1 + s2 + s3) - s3 + s5 = 2s1 + s2 + s5 In conclusion, the green parallelogram and the pink shape (drepanoid) are of the same area. Q.E.D. This peculiar curvilinear shape known as “drepanoid” (from Greek ‘δρέπανον’ drepanon, “sickle”) was probably devised by Greek mathematician Archimedes along with other curves with strange names like “arbelos”, “pelecoid”, and “salinon”. Please spread the word about the Sunday Puzzle!

Text and images by Gianni A. Sarcone 
Labels: SUNDAY PUZZLE, puzzles to solve, recreational mathematics, Archimedian curve
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Sunday Puzzle by Gianni A. Sarcone is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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