Cylinder and Spheres

Introduction

• Cylinders and Spheres are three-dimensional geometric shapes that play essential role in mathematics, architecture, engineering, and everyday life.
• Both shapes can be described using the radius ($r$) as a key measurement, but they differ in shape and symmetry.

What is a Cylinder?

• A Cylinder is a three-dimensional geometric shape with two parallel circular bases that are congruent (same size and shape).
• These Bases are connected by a curved surface.

Key Features of a Cylinder:

• Bases – Two circular and parallel faces.
• Height ($h$) – The perpendicular distance between the two bases.
• Radius ($r$) – The distance from the center to the edge of the circular base.

What is a Sphere?

• A Sphere is a perfectly symmetrical three-dimensional shape where Every point on the surface is the same distance from a central point (called the center).
• It has no edges or vertices.

Key Features of a Sphere:

• Radius ($r$) – The distance from the center to any point on the surface.
• Diameter ($d$) – Twice the radius ($d = 2r$).

How to Find the Volume of Cylinder and Spheres?

• Volume is the amount of space occupied by an object.

• The Formula for its volume is:

• $r$ = radius of the base
• $h$ = height

• The Formula for its volume is:

• $r$ = radius of the base

How to Find the Surface Area of Cylinder and Spheres?

• Surface area is the Total Area of all the surfaces (faces, bases, and curved sides) that cover a 3D object.

• The Formula for its Surface area is:

• $r$ = radius of the base
• $h$ = height

• The Formula for its Surface area is:

• $r$ = radius of the base

Solved Examples

Solved Example

Problem: A cylindrical water tank has a diameter of $14\text{ meters}$ and a height of $5\text{ meters}$. Calculate its volume. (Use $\pi = 3.14$)

SOLUTION

1

Find the radius:

• Given diameter ($d$) = $14\text{ m}$

$$r = \frac{14}{2} = 7\text{ m}$$

2

Put the values in formula:

$$V = \pi(7)^2(5)$$

3

Compute the Result:

$$V = 3.14 \times 49 \times 5 \approx 769.69\text{ m}^3$$

Final Answer: $769.69\text{ m}^3$

Solved Example

Problem: A basketball has a diameter of $24\text{ cm}$. Calculate its volume. (Use $\pi = 3.14$)

SOLUTION

1

Find the radius:

• Given diameter ($d$) = $24\text{ cm}$

$$r = \frac{24}{2} = 12\text{ cm}$$

2

Put the values in formula:

$$V = \frac{4}{3}\pi(12)^3$$

3

Compute the Result:

$$V = \frac{4}{3} \times 3.14 \times 1728 \approx 7238.23\text{ cm}^3$$

Final Answer: $7238.23\text{ cm}^3$

Solved Example

Problem: A pipe has a radius of $5\text{ cm}$, and a height of $12\text{ cm}$. Find its total surface area. (Use $\pi = 3.14$)

SOLUTION

1

Find the radius: Radius of a cylinder is $5\text{ cm}$

2

Put the values in formula:

$$A = 2\pi(5)^2 + 2\pi(5)(12)$$

3

Compute the Result:

$$A = 50\pi + 120\pi = 170\pi \approx 534.07\text{ cm}^2$$

Final Answer: $534.07\text{ cm}^2$

Solved Example

Problem: A globe has a diameter of $30\text{ cm}$. Find its surface area. (Use $\pi = 3.14$)

SOLUTION

1

Find the radius:

• Given diameter ($d$) = $30\text{ cm}$

$$r = \frac{30}{2} = 15\text{ cm}$$

2

Put the values in formula:

$$A = 4\pi(15)^2$$

3

Compute the Result:

$$A = 4\pi(225) = 900 \times 3.14 \approx 2827.43\text{ cm}^2$$

Final Answer: $2827.43\text{ cm}^2$