# Surface Area of a Cylinder

## Summary

A cylinder is a right circular prism. It is a solid object with 2 identical flat circular ends, and a curved rectangular side.

The figure above depicts the developmental figure of a cylinder whose base radius is \(r\) and height is \(h.\) The surface area is equal to the sum of the areas of the two circular bases and the rectangular side. The area of each base is \(\pi r^2.\) Since the width of the rectangular side must be equal to the circumference of the base, the area of the rectangular side is \(2\pi rh.\) Therefore the total surface area is \(2\pi r^2+2\pi rh=2\pi r(r+h).\)

Note: Sometimes, the definition of cylinders may not require having a circular base. In such cases, the base shape will need to be given. The above definition is then called a circular cylinder.

## Examples

## What is the surface area of a cylinder whose base is a circle of radius 3 and height of length 4?

The surface area is \( 2 \pi \times 3 \times 4 +2 \pi \times 3^2 = 42 \pi \). \( _\square \)

## Suppose that the sum of the areas of 2 identical circular ends in a cylinder is the same as the area of the curved side of the cylinder. If the radii of of the flat circular ends are each \(r,\) what is the height of the cylinder?

The sum of the areas of the 2 identical flat circular ends in the cylinder is \(2 \pi r^2.\) The area of the curved side of the cylinder is \(2 \pi r h,\) where \(h\) is the height of the cylinder. Equating these two gives \[2 \pi r^2=2 \pi r h \Rightarrow h=r.\] Thus, the answer is \(r.\) \( _\square \)

## Suppose the surface area of a circular cylinder with height \(h\) and base radius \(r\) is half the surface area of a circular cylinder with height \(5h\) and base radius \(r.\) What is the ratio \(r:h?\)

From the formula \( 2 \pi r h +2 \pi r^2 \) for the surface area of a circular cylinder, we have the following relation between the two surface areas of interest: \[2 \pi r h +2 \pi r^2=\frac{1}{2}\times\left(2 \pi r \cdot (5h) +2 \pi r^2\right).\] Dividing both sides by \(\pi r\) gives \[\begin{align} 2h+2r&=5h+r\\ r&=3h\\ r:h&=3:1. \ _\square \end{align}\]

## Suppose that the surface area of a circular cylinder is \(20\pi.\) If both the radius \(r\) and height \(h\) of the cylinder are integers and \(r>1,\) what is \(r+h?\)

From the formula \( 2 \pi r h +2 \pi r^2 \) for the surface area of a circular cylinder, we have \[\begin{align} 2 \pi r h +2 \pi r^2&=20 \pi \\ r(h+r)&=10. \qquad (1) \end{align}\] Since \(r>1\) by assumption, if \(r=2,\) then \(h=3.\) Then no other integer value of \(r>2\) satisfies \((1).\) Hence, \[r+h=2+3=5.\]

**Cite as:**Surface Area of a Cylinder.

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