Cylinder Volume and Height Calculation: A Comprehensive Guide

A cylinder is a three-dimensional shape with a circular base and a curved surface that connects the bases. Understanding the relationship between its volume, height, and base diameter is essential for solving various mathematical problems. In this guide, we will walk you through the process of finding the height of a cylinder when the volume and diameter are known.

Understanding the Formula for the Volume of a Cylinder

The formula for the volume of a cylinder is given by:

V πr2h

Where:

V is the volume of the cylinder, π (pi) is the mathematical constant approximately equal to 3.14159, r is the radius of the base of the cylinder, h is the height of the cylinder.

Given Data and Calculations

Let's consider a cylinder with the following given data:

The volume (V) is 1540 cubic centimeters (cm3), The diameter (D) is 14 centimeters (cm).

First, we need to find the radius (r) given the diameter (D). The formula for the radius is:

r D/2

Substituting the given diameter:

r 14 cm / 2 7 cm

Step-by-Step Solution

Now, we can use the volume formula to find the height (h).

Step 1: Area of the Base

The area of the base (A_b) of the cylinder can also be found using the formula for the area of a circle:

A_b πr2

Substituting the radius:

A_b π(7 cm)2 49π cm2

Step 2: Substituting into the Volume Formula

Now, we can rearrange the volume formula to solve for the height:

V πr2h

1540 cm3 π(7 cm)2h

1540 cm3 49π h

To find h, we divide both sides by 49π:

h 1540 cm3 / (49π)

h 1540 / (49 * 3.14159) ≈ 10.002 cm

Verification and Important Points

To verify the result, we can use the same equation by substituting the calculated height back into the volume formula:

1540 cm3 π(7 cm)2 * 10.002 cm

1540 cm3 49π * 10.002 cm

1540 cm3 ≈ 1540 cm3

As expected, the left side and the right side are almost equal, confirming the accuracy of our calculation.

Important Points:

It is always a good practice to check your answers by substituting them back into the original equation to minimize errors. When dealing with real-world applications, it is crucial to use the correct units and constants to ensure precision. Understanding the relationship between the volume, height, and diameter of a cylinder can help in solving more complex problems in engineering, architecture, and physics.

Conclusion

By applying fundamental mathematical principles and the correct formulas, we can easily find the height of a cylinder when given the volume and diameter. This guide provides a detailed, step-by-step solution along with a verification process to ensure the accuracy of the result. Whether you are a student, an engineer, or someone interested in mathematics, mastering these calculations can be incredibly useful in various fields.