Q4 of 18 Page 217

In regular polygons, what is the relation between the number of sides and the degree measure of an internal angle? Can it be stated in terms of proportion?

Let us consider a regular polygon.


We know how to calculate the sum of all its interior angles.


s = 180 (n – 2)


Where, s = sum


N is the number of sides of the regular polygon


Let us use this formula for a triangle:


For a triangle, n = 3


So, s = 180 × (3 – 2)


= 180 × 1


= 180


Let us use this formula for a square:


For a square, n = 4


So, s = 180 × (4 – 2)


= 180 × 2


= 360


Let us use this formula for a pentagon:


For a square, n = 5


So, s = 180 × (5 – 2)


= 180 × 3


= 540


Let us tabulate the results:



From the above table we can see that s/n is not a constant. So, s is not proportional to n.


Let us modify the formula


Let s = 180 × m


Where s = sum


M = (n-2)


N is the number of side of the regular polygon



We can see that s is proportional to m. The constant of proportionality is 180


In ordinary language, we can say this:


The sum of interior angle of a regular polygon is proportional to ‘2 less than the number of sides’.


More from this chapter

All 18 →
1

For squares, is area proportional to square of the length of a side? If so, what is the constant of proportionality?

 2

In rectangles of area one square metre, as the length of one side changes, so does the length of the other side. Write the relation between the lengths as an algebraic equation. How do we say this in the language of proportions?

 3

In triangles of the same area, how do we say the relation between the length of the longest side and the length of the perpendicular from the opposite vertex? What if we take the length of the shortest side instead?

 5

A fixed volume of water is to flow into a rectangular water tank. The rate of flow can be changed by using different pipes. Write the relations between the following quantities as an algebraic equation and in terms of proportions.

i) The rate of water flow and the height of the water level.


ii) The rate of water flow and the time taken to fill the tank.