Volume of a Cylinder: Formulas, Surface Area, and Solved Examples

Volume of a Cylinder: A cylinder is a foundational three-dimensional geometric solid characterized by two parallel, congruent circular bases connected by a smooth, curved surface. Common real-world examples include pipes, storage barrels, and batteries. Understanding the properties of 3D shapes is a cornerstone of geometry; in this article, we break down the formulas for the volume and surface area of a cylinder and provide solved examples to help you master these concepts.

Volume of Cylinder

The volume of a cylinder represents its total internal capacity, quantifying how much material it can hold. Mathematically, this is calculated by multiplying the area of the circular base by the object's total height.

The standard mathematical formula for the volume of a cylinder is defined as:

Volume (V) = Area of base (A) × Height (h)

For any solid cylinder with a base radius 'r' and vertical height 'h', the formula is derived as follows:

Therefore, the Volume of a cylinder (V) = πr²h cubic units.

Surface Area of Cylinder

The surface area of a cylinder includes the curved side surface area plus the area of the two circular flat bases. It can be expressed as:

Total Surface Area = Curved surface area + Area of both circular bases

= (Circumference of base × Height) + (2 × πr²)

= 2πrh + 2πr²

= 2πr(h + r) square units

Volume of Hollow Cylinder

A hollow cylinder is a cylindrical shell with a vacant center, featuring a distinct internal radius and external radius. The cross-section of its base resembles an annular ring. For a hollow cylinder with height 'h', external radius 'R', and internal radius 'r', the thickness is defined as (R - r).

Thus, the volume is determined by:

Volume of a hollow cylinder (V) = π(R² - r²)h cubic units.

Surface area of a hollow cylinder = Curved surface area + Area of both annular bases

= 2πh(R + r) + 2π(R² - r²) square units.

Problems Based on Volume of Cylinder

Q1. Find the volume of a cylindrical water tank with a base radius of 25 inches and a height of 120 inches. (Use π = 3.14)

a) 235,600 cubic inches

b) 235,500 cubic inches

c) 22,570 cubic inches

d) None of the above

Solution:

The given radius is 25 inches.

The given height is 120 inches.

Using the cylinder volume formula:

V = πr²h

V = 3.14 × (25)² × 120

V = 235,500 cubic inches.

The correct answer is (b).

Q2. Calculate the base radius of a cylindrical container with a volume of 440 cm³ and a height of 35 cm. (Use π = 22/7)

a) 4 cm

b) 2 cm

c) 3 cm

d) None of the above

Solution:

Given:

Volume = 440 cm³, Height = 35 cm

Using the formula V = πr²h:

V = πr²h

440 = (22/7) × r² × 35

r² = (440 × 7) / (22 × 35) = 3080 / 770 = 4

Therefore, r = 2 cm.

Q3. A cylindrical pillar is 15 m high with a base diameter of 350 cm. What is the cost of painting the curved surface area at Rs 25 per m²?

a) Rs. 4,125

b) Rs. 4,000

c) Rs. 3,900

d) None of the above

Solution:

The pillar is a right circular cylinder.

Radius = 175 cm = 1.75 m; Height = 15 m.

Curved surface area = 2πrh

= 2 × (22/7) × 1.75 × 15 m²

= 165 m²

Cost = 25 × 165 = Rs 4,125.

Answer (a).

Q4. A cylindrical drainage tile is 21 cm long. If the inner and outer diameters are 4.5 cm and 5.1 cm, respectively, what is the volume of clay required?

Calculate the volume of the material:

a) 6.96π cm³

b) 6.76π cm³

c) 5.76π cm³

d) None of the above

Solution:

Inner radius (r) = 2.25 cm; Outer radius (R) = 2.55 cm

Height (h) = 21 cm

Volume = π(R² - r²)h

= π × (2.55² - 2.25²) × 21

= π × (6.5025 - 5.0625) × 21

= 30.24π cm³

The correct answer is (d).

Q5. What is the height of a solid cylinder with a 5 cm radius and a total surface area of 660 cm²?

a) 10 cm

b) 12 cm

c) 15 cm

d) 16 cm

Solution:

Total surface area = 2πr(h + r) = 660

2 × (22/7) × 5 × (h + 5) = 660

h + 5 = 660 × 7 / 220 = 21

h = 16 cm

Answer (d).

Q6. A cylindrical tank with a 7 m diameter contains water to a depth of 4 m. What is the total area of the wetted surface?

a) 110 m²

b) 126.5 m²

c) 131.5 m²

d) 136.5 m²

Solution:

Wetted surface = (2πrh) + (πr²)

Radius = 3.5 m, Height = 4 m

Area = πr(2h + r)

= (22/7) × 3.5 × (8 + 3.5)

= 11 × 11.5 = 126.5 m²

Answer (b).

Ans: Volume of a cylinder = πr²h cubic units.

Ans: Volume of a hollow cylinder = π(R² - r²)h cubic units.

Ans: Total surface area of a cylinder = 2πr(h + r) square units.

= 126.5 m²

Answer (b)