To find the surface area of a sphere with radius [tex]\( r \)[/tex], we use a specific formula derived from geometric principles.
### Step-by-Step Solution:
1. Understanding the Problem:
We need to determine which mathematical expression corresponds to the surface area of a sphere.
2. Recalling the Formula:
The surface area [tex]\( A \)[/tex] of a sphere is given by the formula:
[tex]\[
A = 4 \pi r^2
\][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere, and [tex]\( \pi \)[/tex] (pi) is a mathematical constant approximately equal to 3.14159.
3. Analyzing the Options:
- Option A: [tex]\( 4 \pi r^2 \)[/tex]
- Option B: [tex]\( 4 \pi r^3 \)[/tex]
- Option C: [tex]\( \frac{4}{3} \pi r^2 \)[/tex]
- Option D: [tex]\( \frac{4}{3} \pi r^3 \)[/tex]
4. Surface Area of a Sphere:
- The formula for the surface area of a sphere is [tex]\( 4 \pi r^2 \)[/tex].
- This formula indicates that the total outer surface area of a sphere is four times the product of pi ([tex]\( \pi \)[/tex]) and the square of the radius ([tex]\( r^2 \)[/tex]).
5. Validating the Correct Option:
- Option A: [tex]\( 4 \pi r^2 \)[/tex] matches the formula for the surface area of a sphere.
- Other Options:
- Option B: [tex]\( 4 \pi r^3 \)[/tex] is incorrect because it implies a volume-related term with a cubic power.
- Option C: [tex]\( \frac{4}{3} \pi r^2 \)[/tex] is incorrect because it's not related to surface area; it's part of the volume formula but missing the cubic power.
- Option D: [tex]\( \frac{4}{3} \pi r^3 \)[/tex] is the formula for the volume of a sphere, not the surface area.
6. Conclusion:
Therefore, the correct expression that gives the surface area of a sphere with radius [tex]\( r \)[/tex] is:
[tex]\[
\boxed{4 \pi r^2}
\][/tex]
