Geometry - Cylinders | Practice Hub

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.

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 volume of a hollow cylinder with an outer diameter of , an inner diameter of and a length of ?

$270\pi ; in^{3}$

$300\pi ; in^{3}$

$288\pi ; in^{3}$

$250\pi ; in^{3}$

$320\pi ; in^{3}$

Explanation

The general formula for the volume of a hollow cylinder is given by $V=\pi (r_{o}^{2}-r_{i}^{2})l$ where $r_{o}$ is the outer radius, $r_{i}$ is the inner radius, and is the length.

The question gives diameters and we need to convert them to radii by cutting the diameters in half. Remember, . So the equation to solve becomes:

$V=\pi (4^{2}-1^{2})18$ or $V=270\pi; in^{3}$

What is the volume of a hollow cylinder with an outer diameter of , an inner diameter of and a length of ?

$270\pi ; in^{3}$

$300\pi ; in^{3}$

$288\pi ; in^{3}$

$250\pi ; in^{3}$

$320\pi ; in^{3}$

Explanation

The general formula for the volume of a hollow cylinder is given by $V=\pi (r_{o}^{2}-r_{i}^{2})l$ where $r_{o}$ is the outer radius, $r_{i}$ is the inner radius, and is the length.

The question gives diameters and we need to convert them to radii by cutting the diameters in half. Remember, . So the equation to solve becomes:

$V=\pi (4^{2}-1^{2})18$ or $V=270\pi; in^{3}$

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 =

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

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

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 =

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

A cylinder is cut in half and placed on top of a rectangular prism as shown by the figure below.

Explanation

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

From the figure, you should notice that the length of the prism is also the diameter of the half cylinder. Thus, half the length of the prism will also be the radius of the half cylinder. Also, notice that the width of the prism will be the height of the cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{10}{2}=5$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (5)^2(20)}{2}=250\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(10)(20)(8)=1600$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=250\pi+1600=2385.40$

Remember to round to places after the decimal.

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.

$\text{Surface Area of Cylinder}=2\pi(11)^2+2\pi(11)(20)=682\pi=2142.57$

Make sure to round to places after the decimal point.

A cylinder is cut in half and placed on top of a rectangular prism as shown by the figure below.

Explanation

In order to find the volume of the figure, we will first need to find the volumes of the rectangular prism and the half cylinder.

For the given cylinder then, find the radius.

$\text{radius}=\frac{\text{length}}{2}=\frac{10}{2}=5$

Then, recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi r^2 h$ , where is the radius and is the height.

Divide the volume by to find the volume of the half cylinder.

$\text{Volume of Half Cylinder}=\frac{\pi r^2 h}{2}$

$\text{Volume of Half Cylinder}=\frac{\pi (5)^2(20)}{2}=250\pi$

Next, recall how to find the volume of a rectangular prism.

$\text{Volume of Rectangular Prism}=\text{length}\times\text{width}\times\text{height}$

Plug in the given values to find the volume of the rectangular prism.

$\text{Volume of Rectangular Prism}=(10)(20)(8)=1600$

To find the volume of the figure, add the volume of the half cylinder and the volume of the rectangular prism together.

$\text{Volume of Figure}=\text{Volume of Half Cylinder}+\text{Volume of Rectangular Prism}$

$\text{Volume of Figure}=250\pi+1600=2385.40$

Remember to round to places after the decimal.

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.

$\text{Surface Area of Cylinder}=2\pi(2)^2+2\pi(2)(15)=68\pi=213.63$

Make sure to round to places after the decimal point.

Cylinders

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