A piece of wire when bent to form a circle will have a radius of 84 cm. If the wire is bent to form a square, the length of a side of the square is

Options

A. 152 cm
B. 132 cm
C. 168 cm
D. 225 cm

Answer

To solve the problem step by step, we need to find the length of a side of the square formed by bending the wire that originally formed a circle with a radius of 84 cm.

### Step 1: Calculate the circumference of the circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Given that the radius \( r = 84 \) cm, we can substitute this value into the formula.

\[ C = 2 \times \pi \times 84 \]

### Step 2: Calculate the numerical value of the circumference
Using the approximate value of \( \pi \approx 3.14 \):
\[ C = 2 \times 3.14 \times 84 \]
\[ C = 6.28 \times 84 \]
\[ C = 527.52 \text{ cm} \]

### Step 3: Understand that the length of the wire remains constant
When the wire is bent to form a square, the length of the wire remains the same as the circumference of the circle. Therefore, the perimeter \( P \) of the square is equal to the circumference of the circle.

\[ P = 527.52 \text{ cm} \]

### Step 4: Relate the perimeter of the square to the length of its sides
The perimeter \( P \) of a square is given by the formula:
\[ P = 4a \]
where \( a \) is the length of one side of the square. We can set this equal to the circumference we calculated.

\[ 4a = 527.52 \]

### Step 5: Solve for the length of one side of the square
To find \( a \), we divide both sides of the equation by 4:
\[ a = \frac{527.52}{4} \]
\[ a = 131.88 \text{ cm} \]

### Conclusion
The length of a side of the square is approximately \( 131.88 \) cm.

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