Understand the definition of a diagonal. A diagonal consists of a line segment connecting two angles of a polygon. These angles cannot be adjacent; otherwise, the diagonal would be a side. For instance, consider a baseball diamond, which is technically a square. Two diagonals can be drawn in a baseball diamond: one connecting the first and third bases, and one connecting home plate to second base. These diagonals divide the baseball diamond into four triangular regions, centered roughly around the pitcher's mound; for example, one triangle has vertices at home plate, first base and the pitcher's mound, another has vertices at first base, second base and the pitcher's mound, and so on.
Count the number of sides of the polygon and substitute this number for "s" in the expression (1/24)(s - 1)(s - 2)(s2 - 3s + 12). This expression will give the number of regions of any size regular convex polygon, no matter how many sides it possesses. Continuing with the baseball diamond example, since this polygon is four-sided, replace each "s" with a "4," rendering (1/24)(4 - 1)(4 - 2)(42 - 3*4 + 12).
Simplify this expression according to the order of operations. Within each set of parentheses, first simplify exponents, then perform multiplication, and lastly perform addition and subtraction. The example first becomes (1/24)(3)(2)(16 - 12 + 12), which then turns into (1/24)(3)(2)(16). Multiply all terms inside the parentheses, or alternatively, multiply the second, third and fourth terms, then divide by 24. Doing so in the example produces an answer of four. Hence, a four-sided convex polygon such as a baseball diamond possesses four regions. This algebraic solution mirrors the visual, geometric solution described in Step 1.