π Day: How to visualize π

We know π is related to a circle. Let's see how.

First, consider a circle that has a diameter 1. Our formula is the circumference equals π times the diameter. Since the diameter is 1, we know the circumference is π.

Here's an animation that shows the circumference straightened out to a straight line so we can visualize the distance for this circle's circumference is π, or about 3.14.

Now let's consider circles of diameters 1, 2, and 3 and draw the circumference out as a straight line for each.

Does it appear that the circumference for a circle with a diameter of 2 is twice as large as a circumference of a circle with a diameter of 1?

Does it appear that the circumference for a circle with a diameter of 3 is three times as large as a circumference of a circle with a diameter of 1?

What this means is that there is a linear relationship between the diameter of a circle and the circumference of a circle.

To see that, let's graph the circumference of a circle based upon the diameter. In other words, this graph takes any input diameter and the output is the circumference. As you can see, this relationship is a straight line.

But we know that because the formula is:

$C=\pi \times D$

But if we write this with our usual x/y notation and substitute a common approximation for π, we would write:

$y = 3.14 \times x$

This is an equation for a line. Recall our formula from high school algebra, the equation for a line is:

$y=m \times x + b$

where $m$ is the slope of the line and $b$ is the $y-$intercept. That means for our circle, the $y-$intercept is 0 which means when the diameter is 0, the circumference is 0.

Then the slope of the line is π.

If we return to our previous graph, we can see this linear relationship with circles that have a diameter of 1,2, and 3. Not only do we see that in the line we draw as the circumference, but notice we can create a triangle for each circle.

Each triangle is similar to each other. A similar triangle means they have the same shape but different size. This shows the common proportionality of a circle and a triangle.

When we create a triangle from a circle like this, we can see that both the triangle and circle retain their same shape as they increase in size. Circle's, of course, must always retain the same shape as the diameter changes. But triangles can take on many different forms. But similar triangles are related or proportional to each other.