To find the surface area of a sphere, we need to use the standard formula. The problem provides the diameter of the sphere and asks for its surface area.

Sphere Surface Area Formula

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 and $\pi$ (pi) is a mathematical constant approximately equal to 3.14159 or commonly 22/7 for calculations.

Calculate Sphere Radius

The question gives us the diameter ($d$) of the sphere, which is 112 cm. The radius ($r$) is half of the diameter.

$$ r = \frac{d}{2} $$

Substituting the given diameter:

$$ r = \frac{112 \text{ cm}}{2} $$

$$ r = 56 \text{ cm} $$

Calculate Sphere Surface Area

Now we can substitute the calculated radius into the surface area formula. We will use the approximation $\pi \approx \frac{22}{7}$ as it often leads to cleaner results when the radius or diameter is a multiple of 7.

$$ A = 4 \pi r^2 $$

$$ A = 4 \times \frac{22}{7} \times (56 \text{ cm})^2 $$

First, calculate the square of the radius:

$$ (56 \text{ cm})^2 = 56 \times 56 \text{ cm}^2 = 3136 \text{ cm}^2 $$

Now, substitute this back into the formula:

$$ A = 4 \times \frac{22}{7} \times 3136 \text{ cm}^2 $$

We can simplify the calculation by dividing 3136 by 7:

$$ \frac{3136}{7} = 448 $$

So the equation becomes:

$$ A = 4 \times 22 \times 448 \text{ cm}^2 $$

Multiply the numbers:

$$ A = 88 \times 448 \text{ cm}^2 $$

Performing the multiplication:

$$ 88 \times 448 = 39,424 $$

Therefore, the surface area of the sphere is:

$$ A = 39,424 \text{ cm}^2 $$

Conclusion

The calculated surface area matches one of the provided options. The surface area of a sphere with a diameter of 112 cm is 39,424 cm².