Thomas Greenslade wrote a very simple article for the Physics Teacher in March 2011 sharing a beautiful, wooded model from the L E Knott Apparatus Company which illustrates, at a glance, the principle that the area of a circle is really 1/2 x radius x circumference.

Just look at the circle above. It’s certainly not obvious to me that the area of that circle is 1/2 the radius x the circumference. How obscure a relationship.

Of course, knowing the formulas for the area of a circle and for its circumference in terms of its radius, we can verify this relationship. The area of a circle as p r^2, which is 1/2 x r x 2 p r. So the result using wooden models is exactly what we expect, but this wooden model kit does such a great job of showing in one visceral moment, the motivation for this simple relationship.

In this model, a circle is divided into 12 sections. When unwrapped, its clear each section is a triangle and the area of all the triangles is 1/2 the height, which is the radius x the width, which is the length of the circumference of the circle.

For a finite size section, this is only an approximation, as the shape of each wedge is not a perfect triangle, but has a curved edge. However, this is a great opportunity to exercise your imagination to see that as the number of sections increases and the wedge size decreases, the edge looks more and more flat and this becomes a better and better approximation.

So simple and beautiful a kit illustrates this very fundamental principle very well.