The problem asks for the formula that calculates the lateral area of a right cylinder given its height [tex]\( h \)[/tex] and radius [tex]\( r \)[/tex].
The lateral area (LA) of a right cylinder is the surface area of the sides of the cylinder, excluding the areas of the top and bottom circles.
The formula to calculate this is:
[tex]\[ \text{LA} = 2 \pi r h \][/tex]
Explanation:
1. Understanding lateral area:
- The lateral surface area only includes the sides of the cylinder.
- Imagine unrolling the cylindrical side surface into a rectangle.
2. Dimensions of the rectangle:
- The height of the rectangle would be [tex]\( h \)[/tex] (the height of the cylinder).
- The width of the rectangle would be the circumference of the base circle, which is calculated as [tex]\( 2 \pi r \)[/tex].
3. Area of the rectangle:
- The area can be found by multiplying the height by the width of the rectangle.
- Therefore, the lateral area [tex]\( \text{LA} = \text{height} \times \text{circumference} = h \times 2 \pi r \)[/tex].
4. Putting it all together:
- The lateral area becomes [tex]\( \text{LA} = 2 \pi r h \)[/tex].
Therefore, among the given options:
A. [tex]\( \text{LA} = 2 \pi r \)[/tex] is incorrect as it does not include the height.
B. [tex]\( \text{LA} = 2 \pi r^2 \)[/tex] is incorrect as it represents the total area of two circles, not the lateral area.
C. [tex]\( \text{LA} = 2 \pi r^2 + 2 \pi r h \)[/tex] is incorrect as it includes the areas of the top and bottom circles in addition to the lateral area.
D. [tex]\( \text{LA} = 2 \pi r h \)[/tex] is indeed the correct formula for the lateral area of a right cylinder.
So, the correct answer is:
[tex]\[ \boxed{\text{D}} \: \text{LA} = 2 \pi r h \][/tex]
