When “Same Size” Isn’t the Same: Circles, Ellipses, and Dilation

This example demonstrates a common geometric trap: two figures may appear to have the same size (by visual scale or displayed measures), yet their true geometric “size” is different after a transformation. It focuses on circles and ellipses, how dilation (scaling) preserves similarity but changes absolute measurements in different ways, and how the circumcircle/circumscribed circle concept ties a configuration’s size to the circumradius and related quantities. It also connects naturally to integral-calculus ideas by motivating how “size” can be quantified beyond just a single radius—e.g., by area or other measure that would require accumulation/integration to interpret. Construction-wise, the setup centers on points A and B and the workspace pic1: using A and B to define the relevant circle/ellipse objects, applying dilation to obtain transformed shapes, and comparing the resulting measures associated with circumcircles versus ellipses. Suitable for middle to high school geometry lessons on transformations and measurement, and for advanced students studying the meaning of “size” under geometric transforms.