Level 6: Pythagoras and circles

Geometry · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

This level is about measuring what you cannot reach with a ruler. You prove the Pythagorean theorem three different ways and use it for ladders, boxes, whole-number triples and the distance to the horizon. You put it on the coordinate grid as the distance formula. You find out why \(\pi\) is the same for every circle and where \(\pi r^2\) comes from. And you describe straight lines by their slope and equation.

Lesson	You will be able to
1. The Pythagorean theorem: three proofs	Prove \(a^2 + b^2 = c^2\) by rearranging squares, by similar triangles and with Garfield's trapezoid, and use the converse to test for right angles
2. Using Pythagoras	Find missing sides, ladder heights, box diagonals and horizon distances, simplify square roots, and make Pythagorean triples
3. Distance and midpoint	Find the distance and the midpoint between two points, and use them to test triangles and circles
4. Pi and the circumference	Explain why \(\pi\) is the same for every circle, trap it between polygons, and use \(C = \pi d\)
5. Area of a circle, sectors and arcs	Show why the area is \(\pi r^2\), and find the areas of sectors, segments and rings and the lengths of arcs
6. Slope and the equation of a line	Find slopes and equations of lines, and tell parallel and perpendicular lines from their slopes

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