Chapter 4 of seven · Geometry & Algebra
Area
Area, and quadratics. Meets at completing the square, and at Pythagoras.
- Area of a rectangleCounting unit squares, and why base times height counts them for free.
- Half of a rectangleWhy the area of a right triangle is base times height, divided by two.
- Any triangle is half base times heightThe right-triangle formula, extended to every triangle — including the one case a right triangle never raises.
- Area of a parallelogramBase times height, no division by two — a diagonal already did that job.
- Area of a trapeziumTwo triangles, one diagonal, and a height both of them happen to share.
- PythagorasThe area of the two smaller squares always adds up to the area of the largest one.
- The converse of PythagorasIf the squares add up, the angle is a right angle — proved by congruence, not by area.