The difference between the circum ference and the diameter of a circle is 60 cms. The area of the circle is :

Options

A. `425 cm ^(2)`
B. ` 616 cm^(2)`
C. `154 cm^(2)`
D. `308 cm^(2)`

Answer

To solve the problem step by step, we will follow the information given and apply the formulas for circumference, diameter, and area of a circle.

### Step 1: Understand the relationship between circumference, diameter, and radius.
The formulas we need are:
- Circumference (C) = 2πr
- Diameter (D) = 2r

### Step 2: Set up the equation based on the problem statement.
According to the problem, the difference between the circumference and the diameter is 60 cm:
\[ C - D = 60 \]
Substituting the formulas for circumference and diameter:
\[ 2πr - 2r = 60 \]

### Step 3: Factor out the common term.
We can factor out 2r from the left side of the equation:
\[ 2r(π - 1) = 60 \]

### Step 4: Solve for r (the radius).
Now, divide both sides by 2(π - 1):
\[ r = \frac{60}{2(π - 1)} \]
\[ r = \frac{30}{π - 1} \]

### Step 5: Calculate the area of the circle.
The area (A) of a circle is given by the formula:
\[ A = πr^2 \]
Substituting the value of r we found:
\[ A = π\left(\frac{30}{π - 1}\right)^2 \]

### Step 6: Simplify the area expression.
Calculating the square:
\[ A = π \cdot \frac{900}{(π - 1)^2} \]
\[ A = \frac{900π}{(π - 1)^2} \]

### Step 7: Substitute the value of π (approximately 3.14) to find a numerical answer.
Using π ≈ 3.14:
\[ A ≈ \frac{900 \cdot 3.14}{(3.14 - 1)^2} \]
\[ A ≈ \frac{2826}{(2.14)^2} \]
\[ A ≈ \frac{2826}{4.58} \]
\[ A ≈ 617.5 \text{ cm}^2 \]

### Final Answer:
The area of the circle is approximately 617.5 cm².

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