Figure 1-11 shows you how you can determine the area of a trapezium, or of any polygon, by reducing to triangles. The dotted line connecting A and C divides the figure into the triangles ABC and ACD. The area of the trapezium obviously equals the sum of the areas of these triangles.

Figure 1-12 shows how you could cut a disk into 12 equal sectors. Each of these sectors would constitute a triangle, except for the slight curvature of the side that was originally a segment of the circumference of the disk. If this side is considered the base, then the altitude for each triangle equals the radius (r) of the original disk. The area of each triangle, then, equals

and the area of the original disk equals the sum of the areas of all the triangles. The sum of the areas of all the triangles, however, equals the sum of all the bs, multiplied by r and divided by 2. But the sum of all the bs equals the circumference (c) of the original disk. Therefore, the formula for the area of a circle can be expressed as

However, the circumference of a circle equals the product of the diameter times p (Greek letter, pronounced "pi"). p is equal to 3.14159. . . The diameter equals twice the radius; therefore, the circumference equals 2pr. Substituting 2pr for c in the formula

This is the most commonly used formula for the area of a circle. If we find the area of the circle in terms of circumference.