Problem: The sum of the interior angles of a plane convex polygon is a function of the number of its sides. Represent this function analytically. What values can the argument attain?
We know that the sum of the angles of a triangle is 180º.
To obtain the sum of the angles of a quadrilateral, observe that a quadrilateral can be decomposed into two triangles.
Therefore, the sum of the angles is 360º
A pentagon can be viewed as a polygon formed by joining 3 triangles.
Therefore, the sum of the angles is 3*180º = 540º
A hexagon can be viewed as a polygon formed by joining 4 triangles.
Therefore, the sum of the angles is 4 *180º = 720º
Do you observe the pattern?
Great!!! So, you noticed that the sum of the angles of a polygon depends on the minimum number of triangles that can be fitted in it. Absolutely simple, isn’t it?
But wait, before we jump to writing a function, lets think a bit more.
In how many triangles could we decompose the polygon? Does the number of triangles formed depend on the number of sides of the polygon? Yes!! You guessed it right. It does depend on the number of sides. Now lets see, what is the relation between number of sides of the polygon and the minimum number of triangles into which the polygon can be decomposed.
Observe, quadrilateral formed with 4 sides was decomposed into 2 triangles. Pentagon having 5 sides was decomposed into 3 triangles. Hexagon having 6 sides was decomposed into 4 triangles. Which means, every time, the number of triangles formed was 2 less than the total number of sides.
Thus the function for obtaining the sum of the angles is
S(n) = (n – 2)*180º
In the above function, n is the argument. n can take all values except 1 and 2. ( Can you guess why? For clue, check out the figures obtained with n = 1 or 2. Also, see what happens when you put 1 or 2 in S)