A rope of length 60 cm is cut into two pieces. One piece is used to form a rectangle of length 12 cm and width 6 cm. The other piece is bent into a regular hexagon. What is the length of each side of the hexagon ?

Answer

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

### Step 1: Calculate the perimeter of the rectangle
The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times ( \text{length} + \text{width} ) \]
Here, the length is 12 cm and the width is 6 cm.

### Step 2: Substitute the values into the formula
Substituting the values into the perimeter formula:
\[ P = 2 \times (12 \, \text{cm} + 6 \, \text{cm}) = 2 \times 18 \, \text{cm} = 36 \, \text{cm} \]

### Step 3: Calculate the length of the remaining rope
The total length of the rope is 60 cm. After using 36 cm for the rectangle, we find the remaining length:
\[ \text{Remaining length} = 60 \, \text{cm} - 36 \, \text{cm} = 24 \, \text{cm} \]

### Step 4: Determine the perimeter of the hexagon
The remaining piece of rope is bent into a regular hexagon. The perimeter \( P \) of a regular hexagon is given by:
\[ P = 6 \times \text{side} \]
We know the perimeter of the hexagon is equal to the remaining length of the rope, which is 24 cm.

### Step 5: Set up the equation for the side length
Setting the perimeter of the hexagon equal to the remaining length:
\[ 6 \times \text{side} = 24 \, \text{cm} \]

### Step 6: Solve for the length of each side
To find the length of each side, we divide both sides of the equation by 6:
\[ \text{side} = \frac{24 \, \text{cm}}{6} = 4 \, \text{cm} \]

### Final Answer
The length of each side of the hexagon is **4 cm**.

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