Calculating the Diameter and Surface Area of a Cylinder Given Volume and Height

Cylinders are frequently used in various applications, from engineering to everyday household items. Understanding how to calculate its diameter and surface area from the given volume and height is a fundamental skill. This article outlines the steps to determine these values and provides practical examples to help you grasp the concept easily.

Given Data:

Volume of the cylinder: 5175 ft3 Height of the cylinder: 23 ft

Step 1: Calculating the Diameter

The formula for the volume of a cylinder is:

Volume πr2h
Where r is the radius, and h is the height.

By rearranging this formula, we can find the radius:

πr2h 5175 ft3
r2 5175 ft3 / (π * 23 ft) 71.62 ft2

Taking the square root of both sides, we find the radius:

Radius r √71.62 ft ≈ 8.46 ft

To find the diameter, we use the formula:

Diameter 2r 2 * 8.46 ft ≈ 16.9 ft

Step 2: Calculating the Surface Area

The surface area of a cylinder consists of two parts: the lateral surface area and the area of the two circular bases.

Lateral Surface Area (CSA)

The formula for the lateral surface area is:

CSA 2πrh

Plugging in the values for radius and height, we get:

CSA 2π * 8.46 ft * 23 ft ≈ 1221.10 sq. ft

Total Surface Area

The total surface area includes the lateral surface area plus the areas of the two circular bases:

Surface Area CSA 2πr2

First, we calculate the area of one base:

Base Area πr2 π * (8.46 ft)2 ≈ 225.0 sq. ft

Therefore, the total surface area is:

Total Surface Area 1221.10 sq. ft 2 * 225.0 sq. ft ≈ 1671.10 sq. ft

Further Calculations for Precision

For a more precise calculation of the diameter and surface area, we can use more accurate values for π (π ≈ 3.14159) and solve the equations step by step:

Diameter Calculation

The base area is given by:

A 5175 ft3 / 23 ft 225 ft2

Solving for d (diameter) in the formula A (πd2/4), we get:

d2 225 * 4 / π ≈ 4225 / π

d √(4225 / π) ≈ 30 / √π ≈ 16.93 ft

Surface Area Calculation

The lateral surface area (CSA) can be calculated using:

CSA πdh π * 30 / √π * 23 ≈ 1223.3 sq. ft

The total surface area is then:

Total Surface Area CSA 2 * (πd2/4) ≈ 1223.3 sq. ft 2 * 225 ≈ 1673.3 sq. ft

Conclusion

In summary, given the volume and height of a cylinder, we can calculate its diameter and surface area using the formulas for the volume and surface area of a cylinder. This article has demonstrated how to perform these calculations step by step, providing a clear and concise guide for anyone looking to understand these geometric concepts better.

Related Keywords

cylinder volume cylinder diameter cylinder surface area