SOLUTION: a circle is circumscribed around a square. a square is then circumscribed around the circle. what is the ratio of the area of the smaller square to the area of the larger square?
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Question 178880: a circle is circumscribed around a square. a square is then circumscribed around the circle. what is the ratio of the area of the smaller square to the area of the larger square? Answer by solver91311(24713) (Show Source):
Since the size of a unit can be selected arbitrarily, the radius of the circle can be considered to be 1 unit without loss of generality.
Since the radius of the circle is 1 unit, and the vertices of the inscribed square lie on the circle, the diagonal of the inscribed square is 2 units.
The diagonal of a square forms the hypotenuse of an isoceles right triangle with two sides of the square. Therefore, a square with a diagonal that measures 2 units has a side measure of . The area of such a square is .
Since the circle is tangent to all of the sides of the circumscribed square, the measure of a diameter of the circle is the measure of the distance between two opposite sides of the circumscribed square. This measure is equivalent to the measure of the side of the square.
The diameter of a circle with radius 1 is 2, therefore the circumscribed square has a side that measures 2. The area of such a square is .
Therefore, the ratio of the area of the small square to the large square is 1:2.
