To find the lateral area of a right cylinder where [tex]\( h \)[/tex] is the height and [tex]\( r \)[/tex] is the radius, we can follow these steps:
1. Understanding the Lateral Area of a Cylinder:
The lateral area (LA) of a right cylinder is the area of the curved surface that wraps around the cylinder. It does not include the area of the top and bottom circular bases.
2. Relationship Between Dimensions:
You can imagine "unrolling" the lateral surface of the cylinder to form a rectangle. The height [tex]\( h \)[/tex] of the cylinder corresponds to one dimension of the rectangle, while the circumference of the base of the cylinder corresponds to the other dimension.
3. Calculating the Circumference:
The circumference [tex]\( C \)[/tex] of the circular base of the cylinder is given by:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base.
4. Forming the Rectangle:
When you unroll the lateral surface, the length of the rectangle will be the circumference of the base [tex]\( 2 \pi r \)[/tex], and the width of the rectangle will be the height of the cylinder [tex]\( h \)[/tex].
5. Calculating the Lateral Area:
The lateral area (LA) can be found by multiplying the circumference of the base by the height of the cylinder:
[tex]\[
LA = \text{circumference} \times \text{height} = (2 \pi r) \times h = 2 \pi r h
\][/tex]
6. Identifying the Correct Formula:
We see that the correct formula for the lateral area of a right cylinder is:
[tex]\[
LA = 2 \pi r h
\][/tex]
7. Comparing Options:
- Option A: [tex]\( LA = 2 \pi r \)[/tex] (Incorrect, as it does not include the height)
- Option B: [tex]\( LA = 2 \pi r h \)[/tex] (Correct, based on our derived formula)
- Option C: [tex]\( LA = \pi r h \)[/tex] (Incorrect, as it misses a factor of 2)
- Option D: [tex]\( LA = 2 \pi r^2 \)[/tex] (Incorrect, as it incorrectly uses [tex]\( r^2 \)[/tex] instead of [tex]\( r \times h \)[/tex])
Given these comparisons, the correct answer is clearly:
B. [tex]\( LA = 2 \pi r h \)[/tex]
