2. Draw a square ABCD with side 3 cm. Rotate the square ABCD clockwise about A as the point
of rotation and find the order of rotational symmetry.
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The square ABCD with 3 cm sides is rotated by 45 degrees keeping the center fixed to result into another square PQRS. What is the area of the region common to both the squares?
The region common to both the squares is a regular octagon. At the 4 corners 4 small right angle triangles are removed.
Each side of ABCD is formed of a + a√2 + a = 3, where a is the side of each right angle triangle and a√2, the hypotenuse. So
a = 3/(1+√2+1) = 3/3.414213562 = 0.878679656 cm.
The total area of the 4 small right angle triangles = 4*0.878679656^2/2 = 1.544155877 sq cm.
The area of the regular octagon common to both the squares, ABCD and PQRS = 9 - 1.544155877 = 7.455844123 sq cm.
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