How to find the surface area of a cylinder

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Math › How to find the surface area of a cylinder

Questions 1 - 10

Find the surface area of the given cylinder.

Explanation

To find the surface area of the cylinder, first find the areas of the bases:

$\text{Area of Bases}=2\pi r^2$

Next, find the lateral surface area, which is a rectangle:

$\text{Lateral Surface Area}=2\pi r h$

Add the two together to get the equation to find the surface area of a cylinder:

$\text{Surface Area of Cylinder}=2\pi r^2+2\pi r h$

Plug in the given height and radius to find the surface area.

$\text{Surface Area of Cylinder}=2\pi(7)^2+2\pi(7)(14)=294\pi=923.63$

Make sure to round to places after the decimal point.

What is the surface area of a cylinder of height in, with a radius of in?

$152\pi$

$122\pi$

$120\pi$

$136\pi$

$240\pi$

Explanation

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * r^2$

For our problem, this is:

$\pi * 4^2 = \pi * 16$

You need to double this for the two bases:

$2 * 16 * \pi = 32 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 4 * 15 = 120 * \pi$

Therefore, the total surface area is:

$32\pi + 120\pi = 152\pi$

Given a cylinder with radius of 5cm and a height of 10cm, what is the surface area of the entire cylinder?

$150\pi ; cm^2$

$100\pi ; cm^2$

$175\pi ; cm^2$

$200\pi; cm^2$

$50\pi; cm^2$

Explanation

The surface area of the whole cylinder = (2 * area of circle) + lateral area

Think of the lateral area as the paper label on a can; It wraps around the outside of the can while leaving the top and bottom untouched. The area of the circle, times 2, is to account for the top and the bottom of the cylinder.

Area of a circle = $\pi r^2$

So the area of the circle = $25\pi$ , and since there are two circles we have

Now for the lateral area. Notice how if we have a can with a paper label, we can take the label, cut it, and unroll it from the can. In this way, our label now looks like a rectangle with a

height = height and the

width = circumference of the circle.

Circumference = $2\pi r$

So our rectangle is going to have a height of 10 and a width of 10 $\pi$ . So the lateral area =

$10\cdot 10\pi =100\pi$

So the total surface area =

$50\pi +100\pi =150\pi \ cm^{2}$

Find the surface area of the following cylinder.

Cylinder

$252 \pi m^2$

$240 \pi m^2$

$260 \pi m^2$

$256 \pi m^2$

$246 \pi m^2$

Explanation

The formula for the surface area of a cylinder is:

$SA = lateral\ area + 2\cdot (base)$

$SA = 2 \pi r (h) + 2 (\pi r^2)$

where is the radius of the base and is the length of the height.

Plugging in our values, we get:

$SA = 2 \pi (7m) (11m) + 2 \pi (7m)^2$

$SA = 154 \pi m^2 + 98 \pi m^2 = 252 \pi m^2$

The base of a cylinder has an area of $9\pi$ and the cylinder has a height of . What is the surface area of this cylinder?

$90\pi$

$18\pi$

$72\pi$

Explanation

The standard equation for the surface area of a cylinder is

$SA=2\pi r^2+2\pi rh$

where denotes radius and denotes height. We've been given the height in the question, so all we're missing is the radius. However, we are able to find the radius from the area of the circle:

$Area=\pi r^2$

We know the area is $9\pi$

$9\pi =\pi r^2$ so

Now that we have both and , we can plug them into the standard equation for the surface area of a cylinder:

$SA=2\pi (3)^2+2\pi (3)(12)=18\pi +72\pi =90\pi$

What is the surface area of cylinder with a radius of 3 and height of 7?

$51\pi$

$21\pi$

$63\pi$

$10\pi$

Explanation

The surface area of a cylinder can be determined by the following equation:

$SA=2\pi rh+\pi r^2$

$SA=(2\pi)(3)(7)+\pi(3^2)$

$SA=42\pi+9\pi=51\pi$

Find the surface area of the following partial cylinder.

Partial_cylinder

$234 \pi m^2 + 240m^2$

$240 \pi m^2 + 240m^2$

$234 \pi m^2 + 234m^2$

$234 \pi m^2 + 200m^2$

$200 \pi m^2 + 240m^2$

Explanation

The formula for the surface area of this partial cylinder is:

$SA = 2(\pi r^2)(part)+2(r)(h)+(2 \pi r)(part)(h)$

where is the radius of the cylinder, is the height of the cylinder, and is the sector of the cylinder.

Plugging in our values, we get:

$SA = 2(\pi r^2)(part)+2(r)(h)+(2 \pi r)(part)(h)$

$SA = 2(\pi (6m)^2)\left(\frac{270}{360}\right)+2(6m)(20m)+(2\pi (6m))\left(\frac{270}{360}\right)(20m)$

$SA = 54 \pi m^2 + 240m^2 +180 \pi m^2$

$SA = 234 \pi m^2 + 240m^2$

Find the surface area of the following partial cylinder.

Cylinder_sector

$\frac{104 \pi m^2}{3} + 80m^2$

$\frac{144 \pi m^2}{2} + 80m^2$

$144 \pi m^2+ 80m^2$

$\frac{80 \pi m^2}{3} + 144m^2$

$\frac{144 \pi m^2}{6} + 80m^2$

Explanation

The formula for the surface area of this partial cylinder is:

$SA = 2(\pi r^2)(part)+2(r)(h)+(2 \pi r)(part)(h)$

Where is the radius of the cylinder, is the height of the cylinder, and is the sector of the cylinder.

Plugging in our values, we get:

$SA = 2(\pi (8m)^2)\left(\frac{60}{360}\right)+2(8m)(5m)+(2 \pi (8m))\left(\frac{60}{360}\right)(5m)$

$SA = \frac{64 \pi m^2}{3} + 80m^2 + \frac{40 \pi m^2}{3}$

$SA = \frac{104 \pi m^2}{3} + 80m^2$

This figure is a right cylinder with radius of 2 m and a height of 10 m.

What is the surface area of the right cylinder (m2)?

Explanation

In order to find the surface area of a right cylinder you must find the area of both bases (the circles on either end) and add them to the lateral surface area. The area of the two circles is easy to find with $\pi r^{2}$ but remember to multiply by 2 for both bases

$2\cdot2^{2}\cdot\pi= 25.14$ .

Next find the lateral area. The lateral area if unrounded would be a rectangle with height of 10 m and length equal to the circumference of the base circles. Thus the lateral area is

$4\cdot\pi\cdot10= 125.66$