## Pi Day – How Big Is Pi?

In your time zone Pi Day may already be over.  But here, it still is March 14, the traditional day to celebrate Pi.  Most kids can rattle off that Pi has the value 3.14 (approximately) and that $\pi$ is used in a formula to find the area of a circle: $area = \pi r^2$ where r is the radius of the circle.

Just because kids can rattle off a formula, and can even use the formula to compute the area for a circle with a 2 foot radius, doesn’t mean they appreciate much about where the formula comes from.

A fairly simple model can show the reasonableness of the formula, and suggest why the value of 3.14 for $\pi$ is at least in the right range.

At the left, we have a circle with radius r.  In the middle, we show the yellow circle in relationship with two squares, the big blue one and the smaller (tilted) green one.  On the right, we draw just the two squares without the circle.

From this right figure, we can see that the total area of the blue square is $4 r^2$, since the big blue square is composed of 4 little blue squares, and each of these little blue squares is a square with sides r.  We can also see that the area of the green square is exactly half of the area of the blue square; after all, the green square is made up of 4 green triangles, each of which is a little blue square cut diagonally in half.  So we have $2 r^2$ for the green square.

From the middle picture, we can see that the area of the yellow circle falls between the area of the green square and the area of the blue square.  The area of the circle therefore lies between $2 r^2$ and $4 r^2$.

So a formula that claims the area of the circle to be roughly $3.14 r^2$ is in line with that!