To determine the surface area of a right cylinder, we need to account for both the area of the bases and the lateral surface area of the cylinder.
A right cylinder has two circular bases and one rectangular side that wraps around these bases.
1. Area of the bases:
Each base of the cylinder is a circle with radius [tex]\( r \)[/tex]. The area of one circle is given by:
[tex]\[
\pi r^2
\][/tex]
Since there are two such bases, the total area of the bases is:
[tex]\[
2 \pi r^2
\][/tex]
2. Lateral surface area:
The lateral surface area is the area of the rectangle that wraps around the circular bases. The height of this rectangle is [tex]\( h \)[/tex] (the height of the cylinder), and the width is the circumference of the base circle, which is [tex]\( 2 \pi r \)[/tex]. Thus, the area of the lateral surface is:
[tex]\[
2 \pi r h
\][/tex]
3. Total Surface Area:
To find the total surface area, we add the area of the bases and the lateral surface area:
[tex]\[
2 \pi r^2 + 2 \pi r h
\][/tex]
Based on this information, we need to identify the correct formula(s) from the given options:
- Option A: [tex]\(\pi r^2 + \pi r h\)[/tex] only includes one base area and half the lateral surface area. This is incorrect.
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]. If [tex]\(BA\)[/tex] represents the area of one base, this formula does not account for both bases correctly. This is incorrect.
- Option C: [tex]\(2 \pi r^2\)[/tex] only considers the areas of both bases but omits the lateral surface area. This is incorrect.
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex] includes both the areas of the two bases and the lateral surface area, which is correct.
- Option E: [tex]\(BA + \pi r^2\)[/tex]. This formula would only account for the area of one base plus an extra [tex]\(\pi r^2\)[/tex], which is incorrect for finding the surface area of a right cylinder that includes both bases and the lateral surface.
Thus, the correct formula for finding the surface area of a right cylinder is given by option:
[tex]\[ \boxed{4} \][/tex]
Which corresponds to option D: [tex]\[ 2\pi r^2 + 2\pi r h \][/tex]
