What is the diameter of the largest circle lying on the surface of a sphere of surface area 616 sq cm?

Options

A. 14 cm
B. 10.5 cm
C. 7 cm
D. 3.5 cm

Answer

To find the diameter of the largest circle lying on the surface of a sphere with a given surface area, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Surface Area of a Sphere**:
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

2. **Set Up the Equation**:
We are given the surface area \( A = 616 \, \text{cm}^2 \). We can set up the equation:
\[ 4\pi r^2 = 616 \]

3. **Substitute the Value of \( \pi \)**:
For simplicity, we can use the approximation \( \pi \approx \frac{22}{7} \). Thus, we can rewrite the equation as:
\[ 4 \times \frac{22}{7} \times r^2 = 616 \]

4. **Simplify the Equation**:
First, calculate \( 4 \times \frac{22}{7} \):
\[ \frac{88}{7} r^2 = 616 \]
Now, multiply both sides by 7 to eliminate the fraction:
\[ 88r^2 = 616 \times 7 \]
Calculate \( 616 \times 7 \):
\[ 616 \times 7 = 4312 \]
So we have:
\[ 88r^2 = 4312 \]

5. **Solve for \( r^2 \)**:
Divide both sides by 88:
\[ r^2 = \frac{4312}{88} \]
Calculate \( \frac{4312}{88} \):
\[ r^2 = 49 \]

6. **Find the Radius \( r \)**:
Take the square root of both sides:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

7. **Calculate the Diameter**:
The diameter \( d \) of the sphere is twice the radius:
\[ d = 2r = 2 \times 7 = 14 \, \text{cm} \]

### Final Answer:
The diameter of the largest circle lying on the surface of the sphere is \( 14 \, \text{cm} \).

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