TSA vs CSA of a Cylinder: What's the Difference?
The Surface and Lateral Area of 3D Figures - Cylinders is a key topic in geometry and mensuration that every student encounters, both in school exams and various competitive tests. Knowing how to quickly calculate TSA (Total Surface Area), CSA (Curved/ Lateral Surface Area), and understand related formulas is essential for solving a wide range of real-world and exam-based mensuration problems.
Understanding Surface Area and Lateral Area of Cylinders
A cylinder is a 3D shape with two parallel circular bases connected by a curved surface. Its surface area is made up of two main parts: the area of its curved (side) surface and the area of its two bases. The lateral surface area (also called curved surface area, CSA) measures just the area around the side, while the total surface area (TSA) is the sum of the area of the curved surface and the bases.
Surface area calculations for cylinders are important for everyday situations like determining the material to make a can or how much paint is needed to cover a pipe. This concept is tested in classes 9, 10, and competitive exams like JEE and NEET. At Vedantu, we help make these calculations simple and understandable for everyone.
Key Formulas for Surface and Lateral Area of Cylinders
Here are the essential formulas used to calculate various areas related to cylinders:
| Area Type | Formula | Where |
|---|---|---|
| Curved/ Lateral Surface Area (CSA/ LSA) | \( 2\pi r h \) | \( r \) = base radius, \( h \) = height |
| Total Surface Area (TSA) | \( 2\pi r (h + r) \) or \( 2\pi r h + 2\pi r^2 \) | \( r \) = base radius, \( h \) = height |
| Base Area (each) | \( \pi r^2 \) | \( r \) = base radius |
| Volume (for reference) | \( \pi r^2 h \) | \( r \) = base radius, \( h \) = height |
Working with Formulas: Step-by-Step Examples
Example 1: Find the Curved Surface Area (CSA) of a Cylinder
Suppose a cylinder has a radius of 4 cm and a height of 10 cm.
- Write down the CSA formula: \( CSA = 2\pi r h \)
- Substitute values: \( 2 \times 3.14 \times 4 \times 10 \)
- Calculate: \( 2 \times 3.14 = 6.28 \); \( 6.28 \times 4 = 25.12 \); \( 25.12 \times 10 = 251.2 \) cm²
So, the curved surface area is 251.2 cm².
Example 2: Find the Total Surface Area (TSA) of the Same Cylinder
- Write TSA formula: \( TSA = 2\pi r (h + r) \)
- Substitute: \( 2 \times 3.14 \times 4 \times (10 + 4) \)
- Simplify brackets: \( (10 + 4) = 14 \)
- Multiply: \( 2 \times 3.14 = 6.28 \); \( 6.28 \times 4 = 25.12 \); \( 25.12 \times 14 = 351.68 \) cm²
So, the total surface area is 351.68 cm².
Example 3: Find the Area Paint Needed to Cover the Side of a Pipe (Open Ended)
For an open cylinder (like a tube), only calculate CSA. If \( r = 5 \) cm and \( h = 20 \) cm:
CSA = \( 2\pi r h = 2 \times 3.14 \times 5 \times 20 = 628 \) cm²
Practice Problems
- A closed cylinder has a radius of 3 cm and height 7 cm. Find its TSA.
- Find the CSA of a cylinder with radius 5 cm and height 12 cm.
- If the TSA of a cylinder is 440 cm² and its height is 10 cm, what is its radius?
- A cylindrical water tank is open at the top. Radius = 8 m, height = 5 m. What is its surface area that needs to be painted?
- A cylinder with base diameter 10 cm and height 15 cm is to be wrapped with label only on its curved part. What is the area required?
Common Mistakes to Avoid
- Confusing radius and diameter: Always use radius in formulas (radius = diameter/2).
- Forgetting the unit: Area is always in square units (cm², m², etc.).
- Using CSA when the question asks for TSA or vice versa. Read carefully!
- Ignoring the base: For open cylinders, do not include base area in TSA.
- Incorrect value for π: Use π ≈ 3.14 or as directed in the problem.
Real-World Applications
The concept of surface and lateral area of cylinders is used in:
- Calculating the amount of paint needed to cover pipes or cylindrical tanks.
- Deciding material required to make cans, drums, and other cylindrical objects.
- Designing labels for bottles and cans (only the curved part).
- Manufacturing pipes, pillars, packaging, and even in architecture.
At Vedantu, we connect formulas to such practical uses, making geometry lessons relevant and engaging.
Related Internal Links
- Volume of Cube, Cuboid, and Cylinder
- Surface Area of Cone
- Cuboid and Cube: Properties and Formulas
- Area of a Circle: Formula & Examples
- 3D Formulas
- Mensuration: Concepts and Applications
In this topic, we explored the formulas for surface and lateral area of a cylinder, learned step-by-step how to apply them, and saw their real-life applications. These concepts are not only essential for board and competitive exams, but they also help in solving everyday geometry problems with confidence. For more support, explanations, and practice, Vedantu offers detailed notes and interactive classes tailored for all levels.
FAQs on Surface and Lateral Area of Cylinders Explained
1. What is the formula for the lateral surface area of a cylinder?
The lateral surface area (LSA) of a cylinder, also known as the curved surface area (CSA), is calculated using the formula 2πrh, where 'r' represents the radius and 'h' represents the height of the cylinder. This formula is crucial for understanding the area of the cylindrical surface excluding the circular bases.
2. What is the difference between TSA and CSA of a cylinder?
The Total Surface Area (TSA) of a cylinder includes the areas of both the curved surface and the two circular bases. The Curved Surface Area (CSA), or lateral surface area, only considers the area of the curved cylindrical part. Therefore, TSA = CSA + 2(Area of base) = 2πrh + 2πr² = 2πr(r+h).
3. What is the formula for the total surface area of a cylinder?
The total surface area (TSA) formula for a cylinder is 2πr(r + h), where 'r' is the radius and 'h' is the height. This formula incorporates both the curved surface area and the areas of the two circular bases.
4. What is the surface area of a 3D cylinder?
The surface area of a 3D cylinder depends on whether you're calculating the total surface area (TSA) or just the curved surface area (CSA). TSA considers the entire surface, including the circular bases, while CSA only calculates the area of the cylindrical part. Remember to use the appropriate formula (2πr(r+h) for TSA and 2πrh for CSA) and appropriate units.
5. What is the formula for surface area in 3D?
There's no single formula for surface area in 3D; it varies depending on the shape. For a cylinder, use 2πr(r+h) (TSA) or 2πrh (CSA). For other 3D shapes like cubes, cuboids, cones, and spheres, you'll need different formulas. Understanding the shape's components is key to choosing the correct formula.
6. What is the TSA and CSA of cylinders?
TSA (Total Surface Area) is the sum of the areas of all surfaces of a cylinder, including the circular bases and the curved surface. CSA (Curved Surface Area) or LSA (Lateral Surface Area) refers only to the area of the curved surface. The formulas are: TSA = 2πr(r+h) and CSA = 2πrh.
7. How to calculate TSA of a cylinder?
To calculate the total surface area (TSA) of a cylinder, use the formula 2πr(r + h). Remember to measure the radius ('r') and height ('h') accurately. Always include the correct units (e.g., cm², m²) in your answer. Understanding the formula's components helps in precise calculation.
8. How is surface area used in optimizing material usage in manufacturing?
In manufacturing, calculating surface area is crucial for optimizing material usage. Knowing the surface area helps determine the amount of material needed for things like packaging (cans, tubes), coating, and painting. Minimizing surface area can reduce material waste and production costs.
9. How does increasing the height affect the TSA of a cylinder?
Increasing the height ('h') of a cylinder directly increases its total surface area (TSA). This is because the formula for TSA is 2πr(r+h); a larger 'h' results in a larger TSA, while keeping the radius constant.
10. Why do we use pi (π) in the surface area formulas of cylinders?
Pi (π) is used because the cylinder's base is a circle, and the circumference of a circle is calculated using 2πr. The curved surface of a cylinder can be imagined as a rectangle formed by unrolling it, and its area involves the circle's circumference. Hence π is inherently part of the surface area calculation.
