Deriving the Radius of a Cylinder from Surface Area and Height

Understanding the relationship between the surface area, height, and radius of a cylinder is essential in many practical applications, from engineering to education. This article will explore how to derive the radius of a cylinder's base in terms of its height and surface area.

The Surface Area of a Cylinder

The surface area of a cylinder can be expressed using the following equation:

A 2πr2 2πrh

Where:

A is the surface area of the cylinder r is the radius of the cylinder's base h is the height of the cylinder

Deriving the Radius from Surface Area and Height

To express the radius r in terms of the surface area A and the height h, we start with the surface area formula and rewrite it for r.

Step 1: Simplifying the Surface Area Equation

First, let's simplify the surface area equation using simple approximations and algebraic manipulations. We will use π ≈ 3.14 for simplicity:

A 2πr2 2πrh can be rewritten as:

A ≈ 2(3.14)r2 2(3.14)rh 6.28r2 6.28rh

Step 2: Rearranging to Solve for r

We need to isolate r. First, we rearrange the equation:

6.28r2 6.28rh - A 0

This is a quadratic equation in the form ar2 br c 0, where:

a 6.28 b 6.28h c -A

The quadratic formula is:

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

Step 3: Substituting the Values into the Quadratic Formula

Substituting a 6.28, b 6.28h, and c -A into the quadratic formula, we get:

r frac{-6.28h pm sqrt{(6.28h)^2 - 4 cdot 6.28 cdot (-A)}}{2 cdot 6.28}

Simplifying further:

r frac{-6.28h pm sqrt{39.4h^2 25.1A}}{12.56}

Since the radius cannot be negative, we take the positive root:

r frac{-6.28h sqrt{39.4h^2 25.1A}}{12.56}

Conclusion

Thus, the radius of the base of a cylinder in terms of its surface area A and height h is: