To determine the correct formula for finding the lateral area of a right cylinder, let's analyze each given option step-by-step:
1. Understanding the Problem:
- We need to find the lateral area of the right cylinder.
- The height of the cylinder is denoted by [tex]\( h \)[/tex].
- The radius of the base of the cylinder is denoted by [tex]\( r \)[/tex].
2. Lateral Area of a Right Cylinder:
- The lateral area is essentially the area of the side surface of the cylinder.
- This can be visualized by "unrolling" the side surface of the cylinder into a rectangle.
3. Formulating the Lateral Area:
- The height of this rectangle is [tex]\( h \)[/tex], which is the same as the height of the cylinder.
- The width of this rectangle is the circumference of the base of the cylinder, which is [tex]\( 2 \pi r \)[/tex].
4. Calculating Lateral Area:
- The area of a rectangle is given by the formula: width [tex]\(\times\)[/tex] height.
- Therefore, the lateral area of the cylinder (the area of the rectangle) is:
[tex]\[
L A = (2 \pi r) \times h = 2 \pi r h
\][/tex]
5. Evaluating the Given Options:
- Option A: [tex]\( L A = \pi r h \)[/tex]. This formula does not account for the full circumference of the base, so it is incorrect.
- Option B: [tex]\( L A = 2 \pi r \)[/tex]. This represents the circumference alone and does not account for the height, so it is incorrect.
- Option C: [tex]\( L A = 2 \pi r^2 \)[/tex]. This formula incorrectly squares the radius.
- Option D: [tex]\( L A = 2 \pi r h \)[/tex]. This formula correctly matches the derived lateral area of the cylinder.
Given the correct analysis and derivation, the right formula for finding the lateral area of a right cylinder from the given options is:
[tex]\[
L A = 2 \pi r h
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
