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Visual Learning

We are hunter-gatherers. That means we have programmed visual skills that often lead our learning processes. As Tom West says, we must take advantage of that learning style.

Most people have been drilled to believe that the area of a circle is given by p r². The area of an ellipse is determined by a mystical formula, p a b, where a and b are just the half lengths of the two blessed axes. This can be calculated with integrals, or it can be embedded graphically before any official calculation is done. Take a look.

Here's the circle x² + y² = 1 shown (twice) with the ellipse (x/3)² + y² = 1, and some blue horizontal bars.

The bars hit the y-axis in the same spots in both plots, but the bars in the second plot are 3 times longer than in the first plot. Given that the area inside the circle x² + y² = 1 measures out to p square units, how do the plots reveal the measurement of the area inside the ellipse (x/3)² + y² = 1?