The last problem of the week established that π is less than 4 and greater than 3 by looking at the ratios of the perimeter and width of a square, and circumference and diameter of a circle. This week we consider the area and radius of a circle.

A comparison of two amounts is called a ratio. The ratio of the area of a circle to its radius squared is called π (pi) and it can be expressed as a quotient  .

Again this explanation can sound like Greek to many students – who also need to memorize and apply, pi ≈ 3.14. Like last week, we offer a problem to guide students (with simple geometry and good questions) to reason for themselves why pi must be less than 4 and in this case more than 2.

Imagine that you had never been told that π ≈ 3.14. Use the squares and circles below to prove that the value of π must be less than 4 and greater than 2. For this exercise, all corners that look square are square. The solution will be posted next week.