# The difference between the interior and the exterior angle of a regular polygon is 100degree . find the number of sides of the polygon. ?

Nov 23, 2015

The polygon has 9 sides

#### Explanation:

What information do we know and how do we use it to model this situation?

$\textcolor{g r e e n}{\text{Let the number of sides be } n}$
color(green)("Let internal angle be "color(white)(.......)A_i
color(green)("Let external angle be "color(white)(.......)A_e
Assumption: External angle less than internal angle $\textcolor{g r e e n}{\to {A}_{e} < {A}_{i}}$

Thus color(green)(A_i - A_e>0 => A_i - A_e=100

Not that $\sum \text{ is: the sum of}$

color(brown)("Known: "underline("Sum of internal angles is")color(white)(..)color(green)((n-2)180))

So $\textcolor{g r e e n}{\sum {A}_{i} = \left(n - 2\right) 180. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right)}$

color(brown)("Known:"underline(" Sum of external angles is")color(white)(..)color(green)(360^0))

So $\textcolor{g r e e n}{\sum {A}_{e} = 360 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(2\right)}$

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$\textcolor{b l u e}{\text{Equation (1) - Equation (2)}}$

$\sum \left({A}_{i} - A e\right) = \left(n - 2\right) 180 - 360$

But also $\sum \left({A}_{i} - A e\right) = \sum \text{difference}$

There are $n$ sides each with a difference of ${100}^{0}$
So $\sum \text{difference} = 100 n$ giving:

$\textcolor{g r e e n}{\sum \left({A}_{i} - A e\right) = 100 n = \left(n - 2\right) 180 - 360. \ldots \ldots \ldots \ldots \ldots . \left(3\right)}$

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$\textcolor{b l u e}{\text{Collecting like terms}}$

$100 n = 180 n - 360 - 360$

$80 n = 720$

$n = \frac{720}{80} = 9$