Understanding Polygon Angles: Ratio of Interior to Exterior

Let's break down how to find the ratio of the interior angle to the exterior angle in a regular polygon with 'n' sides. Regular polygons have equal side lengths and equal angle measures.

First, we need to recall the formulas for the interior and exterior angles of a regular polygon.

Calculating Interior Angle of a Regular Polygon

The sum of the interior angles of a polygon with 'n' sides is given by the formula: \((n-2) \times 180^\circ\). Since a regular polygon has 'n' equal interior angles, the measure of one interior angle is this sum divided by 'n'.

Measure of one interior angle $$ = \frac{(n-2) \times 180^\circ}{n} $$

Calculating Exterior Angle of a Regular Polygon

The sum of the exterior angles of any convex polygon is always \(360^\circ\). For a regular polygon, all exterior angles are equal. Thus, the measure of one exterior angle is \(360^\circ\) divided by the number of sides, 'n'.

Measure of one exterior angle $$ = \frac{360^\circ}{n} $$

Finding the Ratio

Now, we need to find the ratio of the interior angle to the exterior angle. We divide the measure of the interior angle by the measure of the exterior angle:

Ratio $$ = \frac{\text{Interior Angle}}{\text{Exterior Angle}} = \frac{\frac{(n-2) \times 180^\circ}{n}}{\frac{360^\circ}{n}} $$

To simplify this complex fraction, we can multiply the numerator by the reciprocal of the denominator:

Ratio $$ = \frac{(n-2) \times 180^\circ}{n} \times \frac{n}{360^\circ} $$

We can cancel 'n' from the numerator and denominator:

Ratio $$ = \frac{(n-2) \times 180^\circ}{360^\circ} $$

Now, simplify the fraction \(\frac{180^\circ}{360^\circ}\), which is \(\frac{1}{2}\):

Ratio $$ = (n-2) \times \frac{1}{2} $$

Ratio $$ = \frac{n-2}{2} $$

This is the ratio of the interior angle to the exterior angle of a regular polygon with 'n' sides.

Comparing with Options

Let's look at the given options:

• Option 1: \(n\)
• Option 2: \(\frac{{{\rm{n}}\,{\rm{ - }}\,1}}{2}\)
• Option 3: \(\frac{{{\rm{n}}\,{\rm{ - }}\,2}}{2}\)
• Option 4: \(\frac{{{\rm{2}}\left( {{\rm{n}}\,{\rm{ - }}\,2} \right)}}{3}\)

Our calculated ratio is \(\frac{n-2}{2}\), which matches Option 3.

Step-by-Step Solution

1. Identify the formulas for the interior and exterior angles of a regular polygon with n sides.
2. Interior Angle = \(\frac{(n-2)180^\circ}{n}\)
3. Exterior Angle = \(\frac{360^\circ}{n}\)
4. Calculate the ratio: \(\frac{\text{Interior Angle}}{\text{Exterior Angle}}\).
5. Ratio = \(\frac{\frac{(n-2)180^\circ}{n}}{\frac{360^\circ}{n}}\)
6. Simplify the expression: Ratio = \(\frac{(n-2)180^\circ}{n} \times \frac{n}{360^\circ}\)
7. Cancel 'n' and simplify the constants: Ratio = \(\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{n-2}{2}\).
8. Match the result with the given options. The result is \(\frac{n-2}{2}\).

Revision Table: Regular Polygon Angle Ratio

Additional Information: Interior and Exterior Angles

The interior and exterior angles at any vertex of a polygon are supplementary, meaning they add up to \(180^\circ\). We can verify this using the formulas:

Interior Angle + Exterior Angle $$ = \frac{(n-2)180^\circ}{n} + \frac{360^\circ}{n} $$

Combine the fractions:

$$ = \frac{(n-2)180^\circ + 360^\circ}{n} $$

Distribute the \(180^\circ\):

$$ = \frac{180^\circ n - 360^\circ + 360^\circ}{n} $$

The \(-360^\circ\) and \(+360^\circ\) cancel out:

$$ = \frac{180^\circ n}{n} $$

Cancel 'n':

$$ = 180^\circ $$

This confirms that the sum of the interior and exterior angles at a vertex is \(180^\circ\) for any polygon, regular or not. However, the specific formulas \(\frac{(n-2)180^\circ}{n}\) and \(\frac{360^\circ}{n}\) are only for regular polygons because all angles are equal.