Find the surface area of a sphere of diameter 21 cm

Q: Find the surface area of a sphere of diameter 21 cm

To find the surface area of a sphere, we use the formula: \[ \text{Surface Area} = 4\pi r^2 \] where $ r $ is the radius of the sphere. Given that the diameter of the sphere is 21 cm, we can find the radius by dividing the diameter by 2: \[ r = \frac{21}{2} = 10.5 \, \text{cm} \] Now, substituting the radius into the surface area formula: \[ \text{Surface Area} = 4\pi (10.5)^2 \] Calculating $ (10.5)^2 $: \[ (10.5)^2 = 110.25 \] Substituting back into the formula: \[ \text{Surface Area} = 4\pi \times 110.25 \] \[ \text{Surface Area} = 441\pi \] Using the approximation $\pi \approx 3.14$: \[ \text{Surface Area} \approx 441 \times 3.14 \] \[ \text{Surface Area} \approx 1385.94 \, \text{cm}^2 \] Rounding to the nearest whole number, the surface area is approximately 1386 cm$^2$. Therefore, Option 1 (1386 cm$^2$) is the correct answer. **Explanation for Incorrect Options:** - **Option 2 (693 cm$^2$)**: This value is exactly half of the correct surface area. It might result from mistakenly using the diameter instead of the radius in the formula, leading to an incorrect calculation. - **Option 3 (66 cm$^2$)**: This value is significantly lower than the correct surface area. It could result from a severe miscalculation or misunderstanding of the formula, possibly involving incorrect arithmetic or unit conversion. - **Option 4 (462 cm$^2$)**: This value is also incorrect and does not correspond to any logical step in the calculation of the sphere's surface area. It might arise from an error in the calculation process, such as using incorrect values or operations. In conclusion, Option 1 is the correct answer as it accurately reflects the calculated surface area of the sphere using the given diameter.

Question

Accepted Answer

To find the surface area of a sphere, we use the formula:

$$ \text{Surface Area} = 4\pi r^2 $$

where $ r $ is the radius of the sphere. Given that the diameter of the sphere is 21 cm, we can find the radius by dividing the diameter by 2:

$$ r = \frac{21}{2} = 10.5 \, \text{cm} $$

Now, substituting the radius into the surface area formula:

$$ \text{Surface Area} = 4\pi (10.5)^2 $$

Calculating $ (10.5)^2 $:

$$ (10.5)^2 = 110.25 $$

Substituting back into the formula:

$$ \text{Surface Area} = 4\pi \times 110.25 $$

$$ \text{Surface Area} = 441\pi $$

Using the approximation $\pi \approx 3.14$:

$$ \text{Surface Area} \approx 441 \times 3.14 $$

$$ \text{Surface Area} \approx 1385.94 \, \text{cm}^2 $$

Rounding to the nearest whole number, the surface area is approximately 1386 cm$^2$.

Therefore, Option 1 (1386 cm$^2$) is the correct answer.

**Explanation for Incorrect Options:**

- **Option 2 (693 cm$^2$)**: This value is exactly half of the correct surface area. It might result from mistakenly using the diameter instead of the radius in the formula, leading to an incorrect calculation.

- **Option 3 (66 cm$^2$)**: This value is significantly lower than the correct surface area. It could result from a severe miscalculation or misunderstanding of the formula, possibly involving incorrect arithmetic or unit conversion.

- **Option 4 (462 cm$^2$)**: This value is also incorrect and does not correspond to any logical step in the calculation of the sphere's surface area. It might arise from an error in the calculation process, such as using incorrect values or operations.

In conclusion, Option 1 is the correct answer as it accurately reflects the calculated surface area of the sphere using the given diameter.