How Do You Calculate the Volume and Surface Area of a Cylinder?
The concept of cylinder plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From storing liquids in tanks to rolling cans, understanding cylinders helps students solve volume and surface area problems smoothly.
What Is Cylinder?
A cylinder is a 3-dimensional geometric solid with two parallel and congruent circular bases connected by a curved surface. You’ll find this concept applied in areas such as Mensuration, geometry, and real-life objects like pipes, cans, and tanks.
Key Formula for Cylinder
Here’s the standard formula:
| Quantity | Formula | Units |
|---|---|---|
| Curved Surface Area (CSA) | 2πrh | sq. units |
| Total Surface Area (TSA) | 2πr(r + h) | sq. units |
| Volume | πr2h | cu. units |
Cross-Disciplinary Usage
Cylinder is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various practical and theoretical questions.
Cylinder Properties
- 3D solid with two parallel, congruent circular bases
- Curved surface connects the two bases
- Height (h): Perpendicular distance between bases
- Radius (r): Distance from center to edge of the base
- Faces: 3 (2 circular, 1 curved)
- Edges: 2 (where the bases meet the curved part)
- No vertices (corners)
- Axis: line joining the centers of the bases
- Symmetry: Infinite lines of symmetry
How to Derive Cylinder Formulas
- Curved Surface Area (CSA):
Unroll the cylinder's side — it's a rectangle with height h and width equal to the circle's circumference (2πr).
Area = Length × Height = 2πr × h = 2πrh
- Total Surface Area (TSA):
Add areas of top and bottom circles (πr² + πr² = 2πr²) to CSA.
TSA = 2πrh + 2πr² = 2πr(r + h)
- Volume:
Volume = Area of base × Height = πr²h
Step-by-Step Illustration
- Sample Problem: Find the volume and total surface area of a cylinder with radius 3 cm and height 7 cm.
-
Volume:V = πr²h = π × (3)² × 7 = π × 9 × 7 = π × 63 ≈ 3.14 × 63 = 197.82 cm³
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Total Surface Area:TSA = 2πr(r + h) = 2 × π × 3 × (3 + 7) = 2 × π × 3 × 10 = 6π × 10 = 60π ≈ 188.4 cm²
Speed Trick or Vedic Shortcut
When a cylinder's diameter is given, halve to get the radius. For quick calculation: multiply radius × radius × height, then multiply by π (approx. 3.14). Estimation works well for MCQs.
Example Trick: For a cylinder with diameter 8 cm and height 5 cm:
1. r = 8 ÷ 2 = 4
2. 4 × 4 × 5 = 80
3. 80 × 3.14 ≈ 251.2 cm³ (volume)
Tricks like these help you save time in timed competitive exams. Vedantu sessions often offer similar tips so you can answer quickly and avoid falling for common mistakes.
Try These Yourself
- Find the curved surface area of a cylinder of radius 6 cm and height 10 cm.
- If the curved surface area is 132 cm² and height 7 cm, find the radius.
- Write any 3 objects around you shaped like cylinders.
- What is the difference between a prism and a cylinder?
Frequent Errors and Misunderstandings
- Forgetting to square the radius in volume formula
- Mixing up height and diameter
- Using the wrong formula for hollow cylinders
- Incomplete units (not writing cm² or cm³)
- Applying π as 3 instead of 3.14 (unless specified)
Relation to Other Concepts
The idea of cylinder connects closely with cube/cuboid and cone topics. Mastering cylinders helps you approach more advanced shapes and questions in mensuration and 3D geometry.
Classroom Tip
A quick way to remember cylinder formulas is “πr²h for Volume, 2πrh for Curved Surface Area, and 2πr(r+h) for Total Area.” Imagine rolling a paper to form a cylinder—you see the circle (base) and rectangle (side) come together! Vedantu’s teachers often use simple objects in class—like cans or glass jars—to reinforce the concept visually during live classes.
We explored cylinder—from definition, formula, examples, mistakes, and connections to other math shapes. Continue practicing with Vedantu to become confident in solving problems involving cylinders quickly and correctly.
Related Topics on Vedantu
- Volume of Cube, Cuboid and Cylinder – Understand how cylinder formulas compare to other 3D shapes.
- Surface Area of Cylinder – A deep dive into curved and total surface areas with more solved problems.
- Right Circular Cylinder – Learn about the most tested cylinder type in exams.
- Cone, Sphere, Cylinder – See side-by-side comparisons and solve mixed shape questions.
FAQs on Cylinder in Maths – Definition, Formulas, Examples & Applications
1. What is a cylinder in Maths?
In mathematics, a cylinder is a three-dimensional geometric shape characterized by two parallel, identical circular bases connected by a single curved surface. The line segment joining the centers of the two circular bases is called the axis, and the distance between the bases is its height (h). The radius of the circular base is known as the cylinder's radius (r).
2. What are the key properties of a cylinder?
A cylinder has several distinct properties that define its structure:
It has three faces: two flat circular faces (the bases) and one curved surface.
It has two curved edges where the curved surface meets the bases.
A cylinder has zero vertices (corners), as vertices are points where straight edges meet.
The two circular bases are always parallel and congruent (identical in size and shape).
3. What are the main formulas used for a cylinder?
The primary formulas for a cylinder, based on its radius (r) and height (h), are:
Volume (V): The space a cylinder occupies is calculated as V = πr²h.
Curved Surface Area (CSA): This is the area of the curved side only, calculated as CSA = 2πrh.
Total Surface Area (TSA): This is the sum of the curved area and the areas of the two circular bases, calculated as TSA = 2πrh + 2πr² or 2πr(h + r).
4. What are some common examples of cylinders in everyday life?
Cylindrical shapes are very common in our daily surroundings. Some familiar examples include:
Food and drink cans (soda cans, canned vegetables)
Water pipes and tubes
Batteries (like AA or AAA)
Gas cylinders for cooking
Candles and fire extinguishers
Toilet paper rolls
5. Why is it important to distinguish between Curved Surface Area (CSA) and Total Surface Area (TSA)?
The distinction is crucial for practical applications. You use Curved Surface Area (CSA) when you only need to cover or calculate the area of the lateral, curved part. For example, finding the area of a label for a can or the cost of painting a pillar. You use Total Surface Area (TSA) when you need to calculate the area of the entire object, including its top and bottom. For instance, determining the amount of sheet metal required to manufacture a sealed cylindrical container.
6. How is a cylinder different from a prism?
While both are 3D shapes with two parallel and congruent bases, the key difference lies in the shape of their base. A cylinder has circular bases and a single curved lateral surface. In contrast, a prism has polygonal bases (like triangles, squares, or hexagons) and flat, rectangular lateral faces connecting the corresponding edges of the bases. This fundamental difference means a prism has straight edges and vertices, whereas a cylinder does not.
7. What is a 'right circular cylinder' and are there other types?
A right circular cylinder is the most common type discussed in school-level mathematics. In this type, the axis (the line connecting the centers of the bases) is perpendicular (at a 90° angle) to the circular bases. However, there is also an oblique cylinder, where the axis is tilted and not at a right angle to the bases, giving it a slanted appearance. While the volume formula remains the same (base area × perpendicular height), the surface area calculation for an oblique cylinder is more complex.
8. If you cut a cylinder open and lay it flat, what shapes do you get?
This is a great way to understand its surface area. When you unroll a closed cylinder, you get three distinct shapes:
Two circles, which are the top and bottom bases of the cylinder.
One rectangle, which represents the curved surface. The height of this rectangle is the height of the cylinder (h), and its length is equal to the circumference of the circular base (2πr). This is why the curved surface area is 2πr × h.
