Right Circular Cylinder: Surface Area & Volume

A right circular cylinder is a three-dimensional geometric shape. It’s formed by the revolution of a rectangle around one of its sides (the axis). The cylinder has two circular bases that are parallel to each other and connected by a curved surface. The term “right” indicates that the axis of the cylinder is perpendicular to the bases. Key features include the radius (r) of the circular bases, and the height (h) of the cylinder, which is the perpendicular distance between the bases.

Formulae

Here are the key formulae for a right circular cylinder:

• Curved Surface Area (CSA): $CSA = 2\pi rh$ (This is the area of the curved side only)
• Total Surface Area (TSA): $TSA = 2\pi r(h + r)$ or $TSA = CSA + 2\pi r^2$ (This is the area of the curved side plus the area of the two circular bases)
• Volume (V): $V = \pi r^2 h$ (This is the amount of space the cylinder occupies)

Examples

Example-1:

Calculate the curved surface area, total surface area, and volume of a right circular cylinder with a radius of 7 cm and a height of 10 cm.

Solution:

• $CSA = 2 \pi rh = 2 \times \frac{22}{7} \times 7 \times 10 = 440 \, cm^2$
• $TSA = 2\pi r(h + r) = 2 \times \frac{22}{7} \times 7 \times (10 + 7) = 748 \, cm^2$
• $V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 10 = 1540 \, cm^3$

Example-2:

A cylindrical water tank has a volume of $616 \, m^3$ and a height of $4 \, m$. Find the radius of the tank.

Solution:

• We know $V = \pi r^2 h$, so we can rearrange to find r: $r^2 = \frac{V}{\pi h}$
• $r^2 = \frac{616}{\frac{22}{7} \times 4} = \frac{616 \times 7}{22 \times 4} = 49$
• Therefore, $r = \sqrt{49} = 7 \, m$

Common mistakes by students

• Confusing Radius and Diameter: Students often mistakenly use the diameter (d) in place of the radius (r) in the formulas. Remember, $d = 2r$.
• Using Incorrect Units: Forgetting to use consistent units throughout the calculations (e.g., using centimeters for radius and meters for height). Always convert units to be consistent.
• Mixing up CSA and TSA: Not understanding the difference between the curved surface area and the total surface area and using the wrong formula based on the context of the problem.
• Incorrect application of $\pi$ : Some students either don’t remember the value of $\pi$ correctly or use wrong formula to apply it.

Real Life Application

Right circular cylinders are found in many real-world objects and applications:

• Drinking cans and bottles: The shape is efficient for stacking and storing liquids.
• Pipes and tubes: Used for transporting fluids and gases.
• Rollers: In printing and manufacturing.
• Water tanks and storage tanks: Cylindrical shapes are strong and efficient for holding large volumes.
• Many architectural designs: Pillars and columns are often cylindrical.

Fun Fact

The ancient Greeks were fascinated by the cylinder. Archimedes, in particular, made important contributions to understanding its properties and how to calculate its volume and surface area. He even used cylindrical shapes to model various objects in nature.

Recommended YouTube Videos for Deeper Understanding

Q.1 A right circular cylinder has a radius of 7 cm and a height of 10 cm. What is its curved surface area?

Check Solution

Ans: A

Curved Surface Area (CSA) = $2 \pi r h = 2 \times \frac{22}{7} \times 7 \times 10 = 440 cm^2$

Q.2 The volume of a right circular cylinder is $308 cm^3$ and its height is 8 cm. What is its radius?

Check Solution

Ans: A

Volume = $\pi r^2 h$. So, $308 = \frac{22}{7} \times r^2 \times 8$. Solving for $r$, we get $r^2 = \frac{308 \times 7}{22 \times 8} = 12.25$. Therefore, $r = \sqrt{12.25} = 3.5 cm$.

Q.3 If the total surface area of a solid right circular cylinder is $616 cm^2$ and the radius of the base is 7 cm, find the height of the cylinder.

Check Solution

Ans: B

Total Surface Area (TSA) = $2 \pi r (h + r)$. So, $616 = 2 \times \frac{22}{7} \times 7 (h + 7)$. Thus, $616 = 44(h + 7)$. Therefore, $h + 7 = 14$, and $h = 7 cm$.

Q.4 A cylindrical tank has a diameter of 14 m and a height of 5 m. What is the volume of the tank?

Check Solution

Ans: A

Radius, $r = 14/2 = 7 m$. Volume = $\pi r^2 h = \frac{22}{7} \times 7^2 \times 5 = 770 m^3$

Q.5 What is the ratio of the curved surface area to the total surface area of a solid right circular cylinder with base radius ‘r’ and height ‘h’?

Check Solution

Ans: B

CSA = $2 \pi r h$. TSA = $2 \pi r (h + r)$. Ratio = $\frac{2 \pi r h}{2 \pi r (h + r)} = \frac{h}{h+r}$.

Next Topic: Right Circular Cone: Surface Area & Volume