Step 1: Understanding the Concept:
A right circular cylinder is a three-dimensional geometric shape with two parallel circular bases of equal radius connected by a curved surface.
The Curved Surface Area (CSA), also known as the lateral surface area, is the area of just the "side" of the cylinder, excluding the flat top and bottom circular faces.
Key Formula or Approach:
The formula for the Curved Surface Area can be derived by "unrolling" the cylinder.
Imagine cutting the curved side of a cylinder vertically and flattening it out.
It will form a rectangle where:
- The height of the rectangle is the height (\( h \)) of the cylinder.
- The length of the rectangle is the circumference of the circular base of the cylinder (\( 2\pi r \)).
Since the area of a rectangle is \( \text{Length} \times \text{Height} \), the area becomes \( 2\pi r \times h \).
Step 2: Detailed Explanation:
Let's evaluate the mathematical components of the cylinder based on the options provided:
- Radius = \( r \)
- Height = \( h \)
1. Curved Surface Area (CSA): This is the area of the curved part. Formula = \( 2\pi rh \).
2. Base Area: The area of one circular end. Formula = \( \pi r^2 \).
3. Total Surface Area (TSA): This is the sum of the curved area and the two base areas.
Formula = \( 2\pi rh + 2\pi r^2 = 2\pi r(r + h) \).
4. Volume: This is the space enclosed by the cylinder. Formula = \( \text{Base Area} \times \text{Height} = \pi r^2h \).
Looking at the options:
- (A) \( 2\pi r^2 \): This is the area of the two circular bases combined.
- (B) \( \pi r^2h \): This is the volume of the cylinder.
- (C) \( 2\pi rh \): This is the correct formula for the Curved Surface Area.
- (D) \( 4\pi r^2 \): This is the formula for the surface area of a sphere.
Therefore, only option (C) correctly describes the Curved Surface Area.
Step 3: Final Answer:
The formula for the curved surface area of a right circular cylinder is \( 2\pi rh \).