Areas of parallelograms and triangles

Area is the space a flat shape covers, counted in unit squares: how much paint it takes to cover it. A rectangle a by b has area ab.

Parallelogram = base × height. Cut off a triangle from one side and move it to the other: the parallelogram becomes a rectangle with the same base and height (Fig. 6.17). For a very slanted ('thin') one, slide the top along in steps, each keeping the area, until the trick works.

The sides alone are not enough. Tilt a parallelogram with fixed sides and its height drops, so its area drops too.

Triangle = ½ × base × height. A triangle is half of a rectangle (or half of a parallelogram made from two copies of it). For an obtuse triangle, take a big right triangle and subtract a small one.

A median halves the area. A median joins a corner to the midpoint of the opposite side. The two triangles it makes have equal bases and the same height, so equal areas, even though their shapes differ.

Rectangles with perimeter 40: there are infinitely many. The 10 × 10 square has the largest area, 100. There is no smallest: the area can be made as close to 0 as you like.