To find the surface area of a right cylinder, we'll break down the surface area into its components:
1. Understanding the components of the surface area:
- Lateral Surface Area: The surface area of the side of the cylinder, which is the height multiplied by the circumference of the base. This can be given by the formula [tex]\(2 \pi r h\)[/tex].
- Area of the Bases: The area of the two circular bases of the cylinder. Each base has an area of [tex]\(\pi r^2\)[/tex]. Since there are two bases, their total area is [tex]\(2 \pi r^2\)[/tex].
2. Combining these components:
- The total surface area [tex]\(A\)[/tex] of the cylinder is the sum of the lateral surface area and the area of the two bases.
- Mathematically,
[tex]\[
A = \text{(Lateral Surface Area)} + \text{(Area of the two bases)}
\][/tex]
[tex]\[
A = 2 \pi r h + 2 \pi r^2
\][/tex]
Now, let's evaluate the given options based on the understanding of the components:
- Option A: [tex]\(2 \pi r^2\)[/tex]\
This formula only represents the area of the two bases and does not include the lateral surface area. Thus, it is not the total surface area.
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]\
Here, [tex]\(BA\)[/tex] stands for the base area, which is [tex]\(\pi r^2\)[/tex]. If we replace [tex]\(BA\)[/tex] with [tex]\(\pi r^2\)[/tex], the formula becomes:
[tex]\[
\pi r^2 + 2 \pi r h
\][/tex]
This includes both the base area and the lateral surface area, so this formula is valid for finding the total surface area.
- Option C: [tex]\(BA + \pi r^2\)[/tex]\
Similar to the previous option, if [tex]\(BA\)[/tex] is replaced by [tex]\(\pi r^2\)[/tex], this formula becomes:
[tex]\[
\pi r^2 + \pi r^2 = 2 \pi r^2
\][/tex]
This formula results in only the total area of the two bases without including the lateral surface area. Thus, it is not the total surface area.
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]\
This includes the area of the two bases ([tex]\(2 \pi r^2\)[/tex]) and the lateral surface area ([tex]\(2 \pi r h\)[/tex]). This is the exact formula for the total surface area of the cylinder, so this formula is valid.
- Option E: [tex]\(\pi r^2 + \pi r h\)[/tex]\
This formula combines the area of one base ([tex]\(\pi r^2\)[/tex]) with half of the lateral surface area ([tex]\(\pi r h\)[/tex]). It is incomplete and does not represent the total surface area of the cylinder.
Given these evaluations, the formulas that correctly find the surface area of a right cylinder are:
- Option B: [tex]\(BA + 2 \pi r h\)[/tex]
- Option D: [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]
Hence, the correct options are B and D.
