Areas by Dissection

Triangle A P Q is tilted, so its height is awkward to find. Instead, find the easy pieces around it and subtract. The rectangle is ten by six, so its area is sixty. Three right-angled triangles surround the shaded one, and each has its two short sides along the rectangle. Triangle A B P has legs ten and four, so its area is twenty. Triangle P C Q has legs two and five, area five. Triangle Q D A has legs five and six, area fifteen. Together they make forty, so the shaded triangle is sixty minus forty, which is twenty.

All three triangles share the same base of eight, and their tips lie on a line parallel to that base. The height of a triangle is the perpendicular distance from its tip to the base, and that distance is the gap between the parallel lines, six, for all three. So every one of them has area one half times eight times six, which is twenty four. The triangles look very different, but their areas are equal. Competition questions use this all the time: slide the tip along a parallel line and the area does not change.

A triangle whose base is one side of a rectangle, with its tip on the opposite side, is exactly half the rectangle, because it has the same base and the same height. Here is a stronger version. P is any point inside the rectangle. Triangles P A B and P C D both have base twelve. Their heights are the distances from P up to A B and down to C D, and those two heights add to the full height, seven. So together they make one half times twelve times seven, forty two, which is half of eighty four. You do not need to know where P is.

A tilted square sits inside a bigger square, and its corners cut each side into three and four. The big square has side seven, so its area is forty nine. Around the tilted square are four identical right-angled triangles, each with legs three and four, so each has area six and together they make twenty four. The tilted square is forty nine minus twenty four, which is twenty five. In general, with pieces a and b, the area is a plus b all squared, minus two a b, which simplifies to a squared plus b squared. That is the picture behind Pythagoras' theorem.

Triangles A B D and A D C have the same tip, A, and their bases lie on the same line, so they have the same height. That means their areas are in the same ratio as their bases. B D to D C is two to three, so the areas are two to three as well. The whole triangle has area forty, split into five equal parts of eight. Triangle A B D gets two parts, sixteen, and triangle A D C gets three parts, twenty four. A median, which goes to the midpoint, always cuts a triangle in half. Quiz and exercises now.