Another puzzle by Presh Talwalkar.
“Thanks to John H. for the suggestion!
A square is inscribed in a quarter circle such that the outer vertices are on the arc of the quarter circle. If the quarter circle has a radius equal to 1, what is the area of the square?
I am told this was given to 7th grade students (ages 12-13), and I think it is a very challenging problem for that age group. In fact I think it is a good problem for any geometry student.”
See the Square in Quarter Circle for solutions.
Another method.
Extend the diagram by completing the quarter circle into a full circle with four squares in four quadrants. Five congruent squares in a shape of a cross ‘X’ will be visible.
Consider the rectangle formed by three adjacent squares. Its diagonal is same as the diameter of the circle which is 2. The sides of the rectangle are in proportion of 3:1. Using Pythagoras, the side of the square is square root (2/5) and its area is 2/5.