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If the measure of each interior angles of a regular polygon is three times the measure of each exterior angles then find the number of sides of the polygon?​

User Sridhar R
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Let's assume the measure of each exterior angle of the regular polygon is
\displaystyle\sf x.

According to the given information, the measure of each interior angle is three times the measure of each exterior angle. Therefore, the measure of each interior angle is
\displaystyle\sf 3x.

In a regular polygon, the sum of all interior angles is given by the formula
\displaystyle\sf (n-2) * 180, where
\displaystyle\sf n is the number of sides of the polygon.

Since each interior angle measures
\displaystyle\sf 3x, the sum of all interior angles can also be expressed as
\displaystyle\sf n * 3x.

Setting these two expressions equal to each other, we can write the equation:


\displaystyle\sf n * 3x = (n-2) * 180

Now, we can solve for
\displaystyle\sf n, the number of sides of the polygon.

Dividing both sides of the equation by
\displaystyle\sf 3x:


\displaystyle\sf n = \frac{{(n-2) * 180}}{{3x}}

Simplifying:


\displaystyle\sf n = \frac{{60 * (n-2)}}{x}

This equation shows that the number of sides
\displaystyle\sf n is proportional to
\displaystyle\sf (n-2). We need more information, specifically the value of
\displaystyle\sf x, to determine the exact number of sides of the polygon.

User Jagira
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