A rhombus has the same perimeter as the circumference of a circle. If each side of the rhombus measures 33 cm, find the radius of the circle.

Answer

To solve the problem step by step, we will find the radius of the circle given that the perimeter of the rhombus is equal to the circumference of the circle.

### Step 1: Calculate the perimeter of the rhombus
The formula for the perimeter \( P \) of a rhombus is given by:
\[ P = 4a \]
where \( a \) is the length of one side of the rhombus. Given that each side of the rhombus measures 33 cm, we can substitute this value into the formula:
\[ P = 4 \times 33 = 132 \text{ cm} \]

### Step 2: Set the perimeter of the rhombus equal to the circumference of the circle
Since the perimeter of the rhombus is equal to the circumference \( C \) of the circle, we have:
\[ C = 132 \text{ cm} \]

### Step 3: Use the formula for the circumference of a circle to find the radius
The formula for the circumference of a circle is:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. We can set this equal to the value we found for the circumference:
\[ 2\pi r = 132 \]

### Step 4: Solve for the radius \( r \)
To find \( r \), we will rearrange the equation:
\[ r = \frac{132}{2\pi} \]
Now, substituting \( \pi \) with \( \frac{22}{7} \):
\[ r = \frac{132}{2 \times \frac{22}{7}} = \frac{132 \times 7}{44} \]

### Step 5: Simplify the expression
Now we simplify:
\[ r = \frac{924}{44} \]
Dividing both the numerator and the denominator by 44:
\[ r = 21 \text{ cm} \]

Thus, the radius of the circle is \( 21 \text{ cm} \).