Question

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?​

Answer 1

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.