Divide Line Segment

into the Perimeter of a Square and Circumference of a Circle

such that the Square and Circle have Equal Areas

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Alternative statement of the Problem

Suppose you had a piece of ribbon 40 inches long. How would you cut the ribbon into two lenghts so as to form a square with one piece and a circle with the other and have the square and the circle have the same amount of area?

Discussion:

Is this a geometry problem or an algebra problem? Either?

Would it help to model the problem?

What role might estimation or approximation play as you investigate the problem or have students examine it?

How might a spreadsheet be used to search for a solution?

GSP?

Hint: We could let AB = 1, AP = k, and PB = 1 - k. Then look to the ratio k : (1 - k) to locate P.

What does it mean to "Find P?" or "Cut the the ribbon . . . to have . . .?"

Would an animation help? Could you build one?

SOLVE THE PROBLEM