Surface Area of a Cylinder

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

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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.

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

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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?

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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 $r:h?$

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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 $$\begin{aligned} 2h+2r&=5h+r\\ r&=3h\\ r:h&=3:1. \ _\square \end{aligned}$$

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 $r+h?$

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