The curved surface area of a right circular cylinder with a height of 15 cm and a base radius of 7 cm is 660 cm².
What is the Curved Surface Area of a Right Circular Cylinder Whose Height is 15 cm and the Radius of the Base is 7 cm?
The curved surface area (CSA) of a right circular cylinder refers to the area of its lateral surface, excluding the areas of its top and bottom circular bases. This is often visualized as the area you would get if you unrolled the cylinder into a flat rectangle.
Understanding the Curved Surface Area Formula
To calculate the curved surface area of a cylinder, we use the following formula:
CSA = 2πrh
Where:
- π (pi) is a mathematical constant approximately equal to 3.14159 or 22/7.
- r is the radius of the base of the cylinder.
- h is the height of the cylinder.
This formula essentially represents the perimeter of the base (2πr) multiplied by the height (h), which gives the area of the rectangle formed when the cylinder's curved surface is flattened. For more details on cylinder formulas, you can refer to resources like Khan Academy or Math is Fun.
Given Dimensions
For this specific problem, we are provided with the following dimensions:
| Dimension | Value | Unit |
|---|---|---|
| Radius (r) | 7 | cm |
| Height (h) | 15 | cm |
Step-by-Step Calculation of Curved Surface Area
Let's calculate the curved surface area using the given values and the formula:
-
Identify the known values:
- Radius (r) = 7 cm
- Height (h) = 15 cm
- We will use the approximation π ≈ 22/7 for precise calculation, as it often yields integer results for dimensions involving multiples of 7.
-
Substitute the values into the formula:
CSA = 2πrh
CSA = 2 × (22/7) × 7 cm × 15 cm -
Perform the calculation:
- First, cancel out the 7 in the denominator with the radius value:
CSA = 2 × 22 × (7/7) cm × 15 cm
CSA = 2 × 22 × 1 cm × 15 cm - Multiply the remaining values:
CSA = 44 × 15 cm²
CSA = 660 cm²
- First, cancel out the 7 in the denominator with the radius value:
Therefore, the curved surface area of the right circular cylinder is 660 cm².
Distinguishing Curved and Total Surface Area
It's important to understand the difference between the curved surface area and the total surface area of a cylinder.
- Curved Surface Area (CSA): As calculated above, this is only the area of the side "wall" of the cylinder.
- Total Surface Area (TSA): This includes the curved surface area plus the area of both the top and bottom circular bases. The formula for TSA is:
TSA = 2πrh + 2πr² (where 2πr² is the area of the two circular bases)
For this cylinder:
- CSA = 660 cm²
- Area of one base = πr² = (22/7) × (7 cm)² = (22/7) × 49 cm² = 22 × 7 cm² = 154 cm²
- Area of two bases = 2 × 154 cm² = 308 cm²
- TSA = CSA + (Area of two bases) = 660 cm² + 308 cm² = 968 cm²
Practical Applications of Cylinder Surface Area
Understanding the surface area of cylinders has numerous practical applications in various fields:
- Manufacturing and Packaging: Calculating the amount of material needed to produce cylindrical cans, pipes, or containers (e.g., for food, beverages, or industrial goods).
- Construction: Estimating the amount of paint required to cover cylindrical pillars, water tanks, or storage silos.
- Engineering: Designing pipelines, HVAC ducts, or structural components where surface area affects heat transfer, material cost, or structural integrity.
- Architecture: Determining the surface area of decorative columns or elements for finishing and material estimates.
Key Takeaways
- The curved surface area of a cylinder is the area of its lateral surface only.
- The formula for curved surface area is
CSA = 2πrh. - Always ensure consistent units for radius and height in calculations.
- For a cylinder with a radius of 7 cm and a height of 15 cm, the curved surface area is exactly 660 cm².
