Area Beyond the Basics

Rectangles, squares, triangles and circles were handled early in this chapter. This track covers everything else, and almost all of it comes from one move: turning an awkward shape into a shape already solved. A parallelogram is a rectangle that has been pushed over, which is why its area is base times perpendicular height and why its slanting side plays no part. A trapezium has two different parallel sides, so it uses the average of them — a formula that falls out of fitting two copies of the shape together into a parallelogram. Rhombuses and kites leave base and height behind entirely and are measured by their diagonals, because a shape whose diagonals cross at right angles fills exactly half the rectangle drawn round them. Then come the shapes with no formula at all, which need none: cut them into pieces that have one and add. The last tutorial adds a slice of a circle to that toolkit, with a warning — unlike arc length, a sector's area uses the radius exactly as given.