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Problem 519

# Find the length of an arc intercepted by a side of a regular hexagon inscribed in a circle whose radius is 18 in.

Expert verified
The length of the arc intercepted by a side of the regular hexagon inscribed in a circle with a radius of 18 inches is $$6\pi$$ inches.
See the step by step solution

## Step 1: Find the Central Angle of the Polygon

Since we have a regular hexagon, all six central angles are equal. The total angle around the circle is 360 degrees. Therefore, the central angle of a hexagon in degrees is: Central Angle (in degrees) = Total Angle / Number of Sides Central Angle (in degrees) = 360° / 6 Central Angle (in degrees) = 60°

## Step 2: Convert the Central Angle to Radians

The formula for arc length requires the central angle to be in radians. To convert the central angle from degrees to radians, we use the following formula: Radians = (Degrees × π) / 180 Central Angle (in radians) = (60° × π) / 180 Central Angle (in radians) = (π/3)

## Step 3: Calculate the Arc Length

Now that we have the central angle in radians, we can use the arc length formula: Arc Length = Radius * Central Angle (in radians) Arc Length = 18 * (π/3) Arc Length = 6π in. So, the length of the arc intercepted by a side of the regular hexagon is 6π inches.

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