Calculating the Radius of a Cylinder: A Comprehensive Guide

Understanding the relationship between the surface area, radius, and height of a cylinder is a fundamental skill in geometry and is crucial for various real-world applications, from engineering to everyday life. This article delves into the intricacies of calculating the radius of a cylinder when the surface area and height are known.

Introduction

The surface area of a cylinder serves as a comprehensive measure that includes the areas of the circular bases and the lateral surface. Understanding how to calculate the radius of a cylinder using its surface area and height is essential for several purposes, such as designing cylinders or analyzing their physical properties.

Understanding Cylinder Surface Area

The total surface area A of a cylinder is given by the formula:

A 2πr2 2πrh

This formula accounts for the areas of both circular bases and the lateral surface of the cylinder.

Calculating the Radius Given Total Surface Area and Height

Let’s assume the total surface area of a cylinder is given to be (84π) cm2 and the height is 11 cm. To find the radius, we can break down the problem into steps as follows:

Step 1: Set Up the Equation

The equation for the total surface area is:

84π 2πr2 2πr(11)

Divide through by (2π) to simplify:

r2 11r - 42 0

Step 2: Solve the Quadratic Equation

This is a standard quadratic equation, which can be solved using the quadratic formula:

r (frac{-b pm sqrt{b^2 - 4ac}}{2a})

Here, a 1, b 11, and c -42. Substituting these values into the formula, we get:

r (frac{-11 pm sqrt{121 - 4(1)(-42)}}{2(1)})

Simplifying inside the square root:

r (frac{-11 pm sqrt{289}}{2})

r (frac{-11 pm 17}{2})

This gives us two solutions: r (frac{6}{2}) and r (frac{-28}{2})

Since the radius cannot be negative, we accept r 3 cm.

Verification

Substitute (r 3) cm back into the total surface area formula:

A 2π(32) 2π(3)(11) 2π(9) 66π 84π cm2

This confirms that the radius is indeed 3 cm when the total surface area is 84π cm2 and the height is 11 cm.

Alternative Calculation

For a different scenario, if the total surface area of a cylinder is given as (64π) cm2 and the height is 3 cm, we can calculate the radius as follows:

Step 1: Set Up the Equation

Using the same surface area formula:

64π 2πr2 2πr(3)

Divide through by (2π) to simplify:

r2 3r - 32 0

Step 2: Solve the Quadratic Equation

Using the quadratic formula with a 1, b 3, and c -32, we get:

r (frac{-3 pm sqrt{9 - 4(1)(-32)}}{2(1)})

Simplifying inside the square root:

r (frac{-3 pm sqrt{137}}{2})

This gives us two solutions: r (frac{3 11.747}{2}) and r (frac{-3 - 11.747}{2})

Since the radius cannot be negative, we accept r ≈ 4.35 cm.

Conclusion

Understanding the relationship between the surface area, radius, and height of a cylinder is essential for solving various geometric and real-world problems. By applying the quadratic formula and simplifying the surface area formula, we can accurately determine the radius of a cylinder, ensuring that our solutions are both practical and mathematically sound.