# M5.4.2

**Pythagorean relationships**.

### Proving the Pythagorean Theorem

A guided classroom exercise that contributes to the long-term development of creative thinking and analysis.

### Measuring the Earth

How to measure the Earth's diameter with a meter stick, a stopwatch, and some math.

### Square Roots Using a Carpenter's Square

As demonstrated in a PUMAS example from Lin H Chambers, "How Now, Pythagoras", master carpenters regularly make practical use of geometry and, at times, the Pythagorean theorem in their craft. I found this out some time ago when a good friend told me how an old-time carpenter he worked with calculated square roots using a carpenter's square.

### Isoperimetric Geometry

The isoperimetric theorem states that: "Among all shapes with an equal area, the circle will be characterized by the smallest perimeter" which is equivalent to "Among all shapes with equal perimeter, the circle will be characterized by the largest area." The theorem's name derives from three Greek words: 'isos' meaning 'same', 'peri' meaning 'around' and 'metron' meaning 'measure'. A perimeter (= 'peri' + 'metron') is the arc length along the boundary of a closed two-dimensional region (= a planar shape).

### How Now, Pythagoras?

You may think of the Pythagorean theorem as useful only to geeky college freshmen who want to calculate how many steps they are saving by walking across the grass. Recently I learned of a very practical use of this theorem in a very unexpected place.