The volume of a cylinder is found using a specific formula that takes into account the radius (or diameter) of the circular base and the height of the cylinder. Here's a detailed explanation:
Understanding the Cylinder's Volume
A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. To calculate its volume, we need to know the area of the base and the height.
Formulas for Calculating Volume
There are two primary formulas to calculate the volume of a cylinder, depending on whether you know the radius or diameter of the base:
-
Using Radius (r):
- The base of a cylinder is a circle, and the area of a circle is given by πr², where 'r' is the radius and π (pi) is approximately 3.14159.
- To get the volume (V), we multiply the base area by the height (h) of the cylinder.
- Formula: V = πr²h
-
Using Diameter (d):
- The diameter (d) is twice the radius, so d = 2r or r = d/2.
- We can substitute r = d/2 into the first formula.
- V = π(d/2)²h which simplifies to:
- Formula: V = πd²h/4
Step-by-step Calculation using Radius:
- Find the radius (r): This is the distance from the center of the circle to any point on the circumference of the circular base.
- Square the radius (r²): Multiply the radius by itself.
- Multiply by π: Multiply the result from step 2 by π (approximately 3.14159). This calculates the area of the circular base.
- Multiply by the height (h): Multiply the result from step 3 by the height of the cylinder. This gives the volume of the cylinder.
Step-by-step Calculation using Diameter:
- Find the diameter (d): This is the distance across the circle through the center.
- Square the diameter (d²): Multiply the diameter by itself.
- Multiply by π: Multiply the result from step 2 by π (approximately 3.14159).
- Multiply by the height (h): Multiply the result from step 3 by the height of the cylinder.
- Divide by 4: Divide the result from step 4 by 4 to obtain the cylinder's volume.
Practical Examples:
- Example 1 (Using Radius):
- A cylinder has a radius of 5 cm and a height of 10 cm.
- V = πr²h = π (5 cm)² 10 cm = π 25 cm² 10 cm = 250π cm³ ≈ 785.4 cm³
- Example 2 (Using Diameter):
- A cylinder has a diameter of 10 cm and a height of 10 cm.
- V = πd²h/4 = π (10 cm)² 10 cm / 4 = π 100 cm² 10 cm / 4 = 1000π cm³/4 = 250π cm³ ≈ 785.4 cm³
Summary Table:
| Variable | Description | Formula |
|---|---|---|
| r | Radius of the base | V = πr²h |
| d | Diameter of the base | V = πd²h/4 |
| h | Height of the cylinder | - |
| V | Volume of the cylinder | - |
Conclusion
In summary, to find the volume of a cylinder, you use the formula V = πr²h if you know the radius, or V = πd²h/4 if you know the diameter. Always use consistent units (e.g., centimeters for radius/diameter and height) to get the volume in cubic units (e.g., cm³).
