Each interior angle of a regular plygon is `144^(@) ` . Find the interior angle of a regular polygon which has double the number of sides as the first polygon.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the interior angle of the first polygon.
The interior angle of the first polygon is given as \(144^\circ\).

### Step 2: Calculate the exterior angle of the first polygon.
The exterior angle can be calculated using the formula:
\[ \text{Exterior Angle} = 180^\circ - \text{Interior Angle} \]
Substituting the given interior angle:
\[ \text{Exterior Angle} = 180^\circ - 144^\circ = 36^\circ \]

### Step 3: Determine the number of sides of the first polygon.
The number of sides \(n\) of a polygon can be found using the formula:
\[ n = \frac{360^\circ}{\text{Exterior Angle}} \]
Substituting the exterior angle we found:
\[ n = \frac{360^\circ}{36^\circ} = 10 \]
So, the first polygon has 10 sides.

### Step 4: Find the number of sides of the second polygon.
According to the problem, the second polygon has double the number of sides as the first polygon:
\[ \text{Number of sides of the second polygon} = 2 \times 10 = 20 \]

### Step 5: Calculate the exterior angle of the second polygon.
Using the same formula for the exterior angle:
\[ \text{Exterior Angle of second polygon} = \frac{360^\circ}{\text{Number of sides}} \]
Substituting the number of sides:
\[ \text{Exterior Angle} = \frac{360^\circ}{20} = 18^\circ \]

### Step 6: Calculate the interior angle of the second polygon.
The interior angle can be calculated using the formula:
\[ \text{Interior Angle} = 180^\circ - \text{Exterior Angle} \]
Substituting the exterior angle we found:
\[ \text{Interior Angle} = 180^\circ - 18^\circ = 162^\circ \]

### Conclusion
The interior angle of the second polygon is \(162^\circ\).

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