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.
Real life Examples:
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.
Volume of a Cylinder:
- The Formula for its volume is:
$$V = \pi r^2 h$$
Where,
- $r$ = radius of the base
- $h$ = height
Volume of a Sphere:
- The Formula for its volume is:
$$V = \frac{4}{3} \pi r^3$$
Where,
- $r$ = radius of the base
Steps to Calculate Volume for Cylinder and Sphere:
1
Find the radius or height.
2
Put the values in formula.
3
Compute the Result.
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Book a Consultation →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.
Surface area of a Cylinder:
- The Formula for its Surface area is:
$$A = 2\pi r^2 + 2\pi rh$$
Where,
- $r$ = radius of the base
- $h$ = height
Surface area of a Sphere:
- The Formula for its Surface area is:
$$A = 4\pi r^2$$
Where,
- $r$ = radius of the base
Steps to Calculate Surface Area for Cylinder and Spheres:
1
Find the radius or height.
2
Put the values in formula.
3
Compute the Result.
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: $2827.43\text{ cm}^2$
