What is the formula for surface area and lateral area of a cylinder with examples
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 3D Cylinders Explained
1. What is the formula for the lateral surface area of a cylinder?
The formula for the lateral surface area of a cylinder is 2πrh, where r is the radius and h is the height.
- The lateral surface area represents only the curved surface, not the top and bottom bases.
- It can also be written as πdh, where d = 2r is the diameter.
- Example: If r = 4 cm and h = 10 cm, then LSA = 2π(4)(10) = 80π cm².
2. What is the formula for the total surface area of a cylinder?
The total surface area of a cylinder is 2πrh + 2πr², which includes the curved surface and both circular bases.
- 2πrh = lateral (curved) surface area
- 2πr² = area of two circular bases
- Example: If r = 3 cm and h = 5 cm, TSA = 2π(3)(5) + 2π(3²) = 30π + 18π = 48π cm².
3. What is the difference between lateral area and total surface area of a cylinder?
The lateral surface area includes only the curved surface, while the total surface area includes the curved surface plus the top and bottom bases.
- Lateral surface area = 2πrh
- Total surface area = 2πrh + 2πr²
- Total surface area is always greater than lateral surface area because it includes extra circular areas.
4. How do you calculate the surface area of a cylinder step by step?
To calculate the surface area of a cylinder, use the formula 2πrh + 2πr² and substitute the known values of radius and height.
- Step 1: Identify the radius (r) and height (h).
- Step 2: Calculate lateral area = 2πrh.
- Step 3: Calculate area of two bases = 2πr².
- Step 4: Add both results.
- Example: r = 2 cm, h = 7 cm → TSA = 2π(2)(7) + 2π(4) = 28π + 8π = 36π cm².
5. Why is the lateral surface area of a cylinder equal to 2πrh?
The lateral surface area equals 2πrh because the curved surface unwraps into a rectangle with length 2πr and height h.
- When unfolded, the cylinder forms a rectangle.
- Length of rectangle = circumference of base = 2πr.
- Height of rectangle = h.
- Area of rectangle = length × height = 2πr × h = 2πrh.
6. How do you find the lateral area of a cylinder using diameter?
To find the lateral surface area using diameter, use the formula πdh, where d is the diameter.
- Since d = 2r, substitute into 2πrh to get πdh.
- Example: If d = 8 cm and h = 6 cm, LSA = π(8)(6) = 48π cm².
- This form is useful when the radius is not given directly.
7. Can you give an example of finding the total surface area of a cylinder?
Yes, to find the total surface area, apply the formula 2πrh + 2πr² with given values.
- Let r = 5 cm and h = 12 cm.
- Lateral area = 2π(5)(12) = 120π.
- Area of two bases = 2π(25) = 50π.
- Total surface area = 120π + 50π = 170π cm².
8. What units are used for surface area of a cylinder?
The surface area of a cylinder is measured in square units such as cm², m², or in².
- If radius and height are in centimeters, the answer will be in cm².
- Surface area always represents two-dimensional measurement.
- Do not confuse it with volume, which is measured in cubic units.
9. What is the relationship between the surface area and volume of a cylinder?
The surface area measures the outer covering, while the volume measures the space inside the cylinder.
- Surface area formula = 2πrh + 2πr².
- Volume formula = πr²h.
- Surface area is in square units, while volume is in cubic units.
- Both depend on radius and height but measure different properties.
10. What are common mistakes when finding the surface area of a cylinder?
Common mistakes include forgetting to include both bases or confusing radius with diameter in the formulas.
- Using r instead of d (or vice versa) without converting.
- Forgetting the 2πr² term in total surface area.
- Not squaring the radius in base area.
- Writing answers without square units.
- Mixing up surface area formula with volume formula πr²h.
