Let `A_(1),A_(2),A_(3),...A_(12)` are vertices of a regular dodecagon. If radius of its circumcircle is 1, then the length `A_(1)A_(3)` is-

Answer

To find the length \( A_1A_3 \) in a regular dodecagon with a circumradius of 1, we can follow these steps:

### Step 1: Understand the Geometry of the Dodecagon
A regular dodecagon has 12 vertices, and the angles between adjacent vertices can be calculated. The center of the dodecagon is denoted as \( O \).

### Step 2: Calculate the Central Angle
The total angle around point \( O \) is \( 360^\circ \). Since there are 12 vertices, the angle between any two adjacent vertices (central angle) is:
\[ \text{Central Angle} = \frac{360^\circ}{12} = 30^\circ \]

### Step 3: Determine the Angle for \( A_1A_3 \)
To find the length \( A_1A_3 \), we note that \( A_1 \) and \( A_3 \) are separated by two vertices, which means the angle \( A_1OA_3 \) is:
\[ \text{Angle } A_1OA_3 = 2 \times 30^\circ = 60^\circ \]

### Step 4: Use the Law of Cosines
In triangle \( OA_1A_3 \), we can apply the Law of Cosines. Since \( OA_1 = OA_3 = 1 \) (the radius), we have:
\[ A_1A_3^2 = OA_1^2 + OA_3^2 - 2 \cdot OA_1 \cdot OA_3 \cdot \cos(60^\circ) \]
Substituting the values:
\[ A_1A_3^2 = 1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(60^\circ) \]
Since \( \cos(60^\circ) = \frac{1}{2} \):
\[ A_1A_3^2 = 1 + 1 - 2 \cdot \frac{1}{2} = 2 - 1 = 1 \]
Thus, taking the square root:
\[ A_1A_3 = \sqrt{1} = 1 \]

### Conclusion
The length \( A_1A_3 \) is \( 1 \).

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