Geometry - How to find the volume of a cylinder

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 $V=270\pi; in^{3}$

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.

A right cylinder has a diameter of and a height of . What is the volume of this cylinder?

$10 \pi: in ^3$

$40 \pi: in^3$

$20 \pi: in^3$

Explanation

The formula to find the volume of a cylinder is: $V = \pi \cdot r^2 \cdot h$ , where is the radius of the cylinder and is the height of the cylinder.

A good point to start in this kind of a formula-based problem is to ask "What information do I have?" and "What information is missing that I need?"

In this case, the problem provides us with the height of the cylinder and its diameter. We have the component of the equation, but we're missing the component. Can we find out ? The answer is yes! Radius is half of diameter. So in this case, because the diameter is , the radius must be .

Now that we have and , we are ready to solve for the volume after substituting in those values.

$V = \pi \cdot r^2 \cdot h$

$V = \pi \cdot (1)^2 \cdot 10$

$V = \pi \cdot \1 \cdot 10$

$V = 10 \pi: in^3$

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 34^2 \times 40=46240\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 15^2 \times 40=9000\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=46240\pi-9000\pi=37240\pi=116992.91$

Make sure to round to places after the decimal.

Find the volume of a cylinder that has a radius of and a height of .

$2025\pi$

$75\pi$

$1225\pi$

$1775\pi$

Explanation

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\text{Area of base}\times\text{height}$

Since the base of a cylinder is a circle, we can write the following equation:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Substitute in the given values to find the volume.

$\text{Volume of Cylinder}=\pi\times 15^2\times 9$

Solve.

$\text{Volume of Cylinder}=2025\pi$

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 20^2 \times 25=10000\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 8^2 \times 25=1600\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=10000\pi-1600\pi=8400\pi=26389.38$

Make sure to round to places after the decimal.

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

Find the volume of the figure.

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.

$\text{radius}=\frac{\text{length}}{2}=\frac{12}{2}=6$

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.

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

$\text{Volume of Half Cylinder}=\frac{\pi (6)^2(20)}{2}=360\pi$

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.

$\text{Volume of Rectangular Prism}=(12)(20)(14)=3360$

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

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

$\text{Volume of Figure}=360\pi+3360=4490.97$

Remember to round to places after the decimal.

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we will first need to find the volume of both cylinders.

Recall how to find the volume of the cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

Now, use the given radius and height to find the volume of the larger cylinder.

$\text{Volume of Larger Cylinder}=\pi\times 10^2 \times 12=1200\pi$

Next, use the given radius and height to find the volume of the smaller cylinder.

$\text{Volume of Smaller Cylinder}=\pi\times 4^2 \times 12=192\pi$

Subtract the volume of the smaller cylinder from the volume of the larger one to find the volume of the figure.

$\text{Volume of Figure}=1200\pi-192\pi=1008\pi=3166.73$

Make sure to round to places after the decimal.

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

$250\pi ; cm^2$

$200\pi; cm^2$

$300\pi;cm^2$

$100\pi;cm^2$

$500\pi;cm^2$

Explanation

The volume of a cylinder is given by $\pi r^2\cdot height$

Notice how the formula for the volume is defined as the area of a circle times the lateral height of the cylinder. It is as if we are taking little paper circles and stacking them one-by-one until we fill up the entire container.

Plugging in the numbers we get:

$\pi (5)^2\cdot 10 = 250\pi$