a regular polygon B with n sides is surrounded by squares and regular pentagons in an alternating pattern. determine the value of n

In this problem, a regular polygon \( B \) with \( n \) sides is surrounded by alternating squares and regular pentagons.

### Step 1: Analyze the alternating pattern of squares and pentagons
Each vertex of the polygon \( B \) will be connected to either a square or a regular pentagon. Since squares and pentagons alternate around the polygon, half of the vertices will connect to squares, and the other half will connect to pentagons.

### Step 2: Internal angles of the polygon \( B \)
For any regular polygon with \( n \) sides, the internal angle at each vertex is given by the formula:
\[
\text{Internal angle} = \frac{(n-2) \times 180^\circ}{n}
\]

### Step 3: Exterior angles and fitting condition
We know that the sum of the exterior angles for any polygon is \( 360^\circ \). The exterior angle at each vertex of \( B \) is:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]

For the alternating squares and pentagons to fit around each vertex of \( B \), the sum of their exterior angles at each vertex must add up to the exterior angle of \( B \).

- The exterior angle of a square is \( 90^\circ \).
- The exterior angle of a regular pentagon is \( 108^\circ \).

Thus, at each vertex of \( B \), the alternating sum of the exterior angles of a square and a pentagon must equal the exterior angle of \( B \):
\[
\frac{360^\circ}{n} = 90^\circ + 108^\circ = 198^\circ
\]

### Step 4: Solve for \( n \)
Now, solve for \( n \) by equating the exterior angle of \( B \) to \( 198^\circ \):
\[
\frac{360^\circ}{n} = 198^\circ
\]
\[
n = \frac{360^\circ}{198^\circ}
\]
\[
n = \frac{360}{198} \approx 1.818
\]

Since \( n \) must be an integer, we round \( n \) to the nearest whole number, but this doesn't give us a valid polygon structure. Thus, on further inspection the answer

A regular polygon B with n sides is surrounded by squares and regular pentagons in an alternating pattern. Determine the value of n.

Explore how to determine the number of sides (n) for a regular polygon surrounded by alternating squares and pentagons. The solution uses polygonal angle formulas and examines the fitting condition for squares and pentagons. Ideal for students in Grades 9-12.

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