The interior angle of a regular polygon is 19 times the size of the exterior angle.
Calculate the size of the interior angle and the number of sides the polygon has.

Question

The interior angle of a regular polygon is 19 times the size of the exterior angle. Calculate the size of the interior angle and the number of sides the polygon has.

UpStudy AI · Accepted Answer

To determine the size of the interior angle and the number of sides of the regular polygon, let's follow these steps:

### 1. Understanding the Relationship Between Interior and Exterior Angles

For any polygon:

- **Interior Angle (I):** The angle inside the polygon at each vertex.
- **Exterior Angle (E):** The angle formed by one side of the polygon and the extension of an adjacent side.

These two angles are supplementary, meaning:
$$ I + E = 180^\circ $$

Given in the problem:
$$ I = 19E $$

### 2. Calculating the Exterior Angle (E)

Substitute the given relationship into the supplementary angle equation:
$$ 19E + E = 180^\circ $$
$$ 20E = 180^\circ $$
$$ E = \frac{180^\circ}{20} $$
$$ E = 9^\circ $$

### 3. Finding the Interior Angle (I)

Now, use the relationship $$ I = 19E $$:
$$ I = 19 \times 9^\circ $$
$$ I = 171^\circ $$

### 4. Determining the Number of Sides (n)

The sum of all exterior angles of any polygon is always $$ 360^\circ $$. For a regular polygon, all exterior angles are equal, so:
$$ n = \frac{360^\circ}{E} $$
$$ n = \frac{360^\circ}{9^\circ} $$
$$ n = 40 $$

### **Final Answer**

- **Interior Angle:** $$ 171^\circ $$
- **Number of Sides:** 40
The interior angle is 171 degrees, and the polygon has 40 sides.

The interior angle of a regular polygon is 19 times the size of the exterior angle.
Calculate the size of the interior angle and the number of sides the polygon has.

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