Video Segments

Shape and size
Similar triangles
Applications of similarity
Similar polygons and solids
Internal ratios of similar figures
Perimeters of similar figures
Areas of similar figures
Volumes of similar figures
Applications to biology


The opening scene shows various objects from real life having the same shape but not necessarily the same size. The narrator asks ``Can we construct a figure with the same shape as another?'' A triangle is moved to various positions by translating it, rotating it, or flipping it over. They are congruent because they have not only the same shape but also the same size.

To change size without changing shape, scaling is introduced. Scaling multiplies lengths of all line segments by the same number and produces a similar figure. Similarity preserves angles and ratios of lengths of corresponding line segments.

Applications show how Thales might have used similarity to find the height of a column and of a pyramid by comparing lengths of shadows. Another application of similarity explains why the sum of the angles in any triangle is a straight angle.

Similarity is discussed for more general polygons and for three-dimensional objects. Animation shows what happens to perimeters, areas, and volumes under scaling, with illustrations from real life.

Similarity is the basis of all measurement. It reveals the secret of map making and scale drawings, and also explains some aspects of photographic images. Similarity helps explain why a hummingbird's heart beats so much faster than a human heart, and why it is impossible for a small creature such as a praying mantis to become as large as a horse.

return to main Project MATHEMATICS! screen

E-mail your questions to Tom Apostol