Areas of polygons

Areas of polygons can be explored using squared paper. A good sequence is to start with a square (diagram 1 below), then move to a rectangle (diagram 2 below), observing that the areas of those are simply the product of two non-parallel sides. The area of the non-rectangular parallelogram (diagram 3 below) is easily discovered by cutting a triangle from one side and joining it to the opposite side to create a rectangle. This shows that the area of a parallelogram is the product of the base and the vertical height. Cutting a parallelogram on one diagonal creates two congruent shapes and shows that the area of a triangle (diagram 4 below) is a half of the area of the associated parallelogram, that is, a half of the product of the base and the vertical height. Next, the area of a trapezium (diagram 5 below) can be found by cutting the non-parallel sides through the midpoints and rotating them to make a rectangle. This shows that the area of the trapezium is the product of the average length of the two parallel sides and the distance between them.

Areas of other polygons might be found by seeing them as combinations of the polygons mentioned above, or might require a trigonometric approach.