24th March 2006 at 00:00

Take two identical regular pentagons and cut them into four isosceles triangles, as shown. Angle a is 360.

Show how you would use all eight triangles, touching edge-to-edge, to make (a) a pentagon; (b) a hexagon. (Hint: the pentagon and hexagon are not necessarily regular.)


A sash window with a semi-circular top is 1 metre wide. The upper 'half' of the window is lowered by 25cm. What is the area of the crescent-shaped opening?

SOLUTION Pentagon puzzles Each of the two original pentagons is cut into three identical isosceles triangles and one other isosceles triangle. The eight triangles can then be placed edge-to-edge as shown.

SOLUTION Sash window (Age 11-15)The opening is 14 m2, or 2500 cm2. The opening is A + E + C. But E =B, since the semi-circle E + D = B + D. So the opening equals A + B + C, which is a rectangle of width 1 metre and height 25cm, or area 0.25m2.

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