Lesson 12
Volume of Right Prisms
Let’s look at volumes of prisms.
12.1: Three Prisms with the Same Volume
Rectangles A, B, and C represent bases of three prisms.
![Three rectangles, A, B, C. Rectangle A is 1 square by 3 squares. Rectangle B is 2 squares by 2 squares. Rectangle C is 6 squares by 2 squares.](https://cms-im.s3.amazonaws.com/9NHnu8VUgAb8kii27oNEAQUy?response-content-disposition=inline%3B%20filename%3D%227.6.C2.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%277.6.C2.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=798190c0641eb14f5dac341d3e46333210fcd5a5349dc195b507c53e929ac5f3)
- If each prism has the same height, which one will have the greatest volume, and which will have the least? Explain your reasoning.
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If each prism has the same volume, which one will have the tallest height, and which will have the shortest? Explain your reasoning.
12.2: Finding Volume with Cubes
This applet has 64 snap cubes, all sitting in the same spot on the screen, like a hidden stack of blocks. You will always know where the stack is because it sits on a gray square. You can keep dragging blocks out of the pile by their red points until you have enough to build what you want.
![A snap cube sits on a grid, with a red point on one corner.](https://cms-im.s3.amazonaws.com/tqNmU6MTC3qgxkKP2xcR97bv?response-content-disposition=inline%3B%20filename%3D%227-7.7.12.3_Snapcube_copy.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.12.3_Snapcube_copy.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=55eda6dc6c2ed44c06b315b0ba6a2dc0e4dd628fcfbff70f0f8f1fcacc278e6b)
Click on the red points to change from left/right movement to up/down movement.
![Two images, each stacks of snap cubes 2 by 2 by 3. On the first image, arrows indicate left and right motion. On the second image, arrow indicated up and down motion.](https://cms-im.s3.amazonaws.com/jb9fZ7vJDLHy2Pnun3ekYjjw?response-content-disposition=inline%3B%20filename%3D%227-7.7.C1.Image.08.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.C1.Image.08.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4f2c57994b2947c026464343babcad15f23350abcca436ec3fe00d945487488e)
There is also a shape on the grid. It marks the footprint of the shapes you will be building.
- Using the face of a snap cube as your area unit, what is the area of the shape? Explain or show your reasoning.
- Use snap cubes to build the shape from the paper. Add another layer of cubes on top of the shape you have built. Describe this three-dimensional object.
- What is the volume of your object? Explain your reasoning.
- Right now, your object has a height of 2. What would the volume be
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if it had a height of 5?
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if it had a height of 8.5?
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12.3: Can You Find the Volume?
The applet has a set of three-dimensional figures.
- For each figure, determine whether the shape is a prism.
- For each prism:
- Find the area of the base of the prism.
- Find the height of the prism.
- Calculate the volume of the prism.
Is it a prism? | area of prism base (cm2) | height (cm) | volume (cm3) |
---|---|---|---|
- Begin by grabbing the gray bar on the left and dragging it to the right until you see the slider.
- Choose a figure using the slider.
- Rotate the view using the Rotate 3D Graphics tool marked by two intersecting, curved arrows.
- Note that each polyhedron has only one label per unique face. Where no measurements are shown, the faces are identical copies.
- Use the distance tool, marked with the "cm," to click on any segment and find the height or length.
- Troubleshooting tip: the cursor must be on the 3D Graphics window for the full toolbar to appear.
Imagine a large, solid cube made out of 64 white snap cubes. Someone spray paints all 6 faces of the large cube blue. After the paint dries, they disassemble the large cube into a pile of 64 snap cubes.
- How many of those 64 snap cubes have exactly 2 faces that are blue?
- What are the other possible numbers of blue faces the cubes can have? How many of each are there?
- Try this problem again with some larger-sized cubes that use more than 64 snap cubes to build. What patterns do you notice?
12.4: What’s the Prism’s Height?
There are 4 different prisms that all have the same volume. Here is what the base of each prism looks like.
![Four polygons on a grid, A, B, C, D.](https://cms-im.s3.amazonaws.com/2hk65vd2mlnqrpwvtqz5wxs2wxvd?response-content-disposition=inline%3B%20filename%3D%227.6.C2.Image.08.png%22%3B%20filename%2A%3DUTF-8%27%277.6.C2.Image.08.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e26fe8bbc95f414014c7958ac316bac9a3a221ce16946876d7d7889493f0a862)
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Order the prisms from shortest to tallest. Explain your reasoning.
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If the volume of each prism is 60 units3, what would be the height of each prism?
- For a volume other than 60 units3, what could be the height of each prism?
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Discuss your thinking with your partner. If you disagree, work to reach an agreement.
Summary
Any cross section of a prism that is parallel to the base will be identical to the base. This means we can slice prisms up to help find their volume. For example, if we have a rectangular prism that is 3 units tall and has a base that is 4 units by 5 units, we can think of this as 3 layers, where each layer has \(4\boldcdot 5\) cubic units.
![Two images. First, a prism made of cubes stacked 5 wide, 4 deep, 3 tall. Second, each of the layers of the prism is seperated to show 3 prisms 5 wide, 4 deep, 1 tall.](https://cms-im.s3.amazonaws.com/zZEpMqMzw2Av8TkDfd1o4f3Z?response-content-disposition=inline%3B%20filename%3D%227-7.6.C2.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C2.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dda8c5271bd2df8a2d84cbf0e3b15e05ea8c14427e0c04a39cd7dd79e38a1b39)
That means the volume of the original rectangular prism is \(3(4\boldcdot 5)\) cubic units.
This works with any prism! If we have a prism with height 3 cm that has a base of area 20 cm2, then the volume is \(3\boldcdot 20\) cm3 regardless of the shape of the base. In general, the volume of a prism with height \(h\) and area \(B\) is
\(\displaystyle V = B \boldcdot h\)
For example, these two prisms both have a volume of 100 cm3.
![Two prisms. First prism has a triangular base with area 20 centimeters squared, and height 5 centimeters. Second prism has an irregular base with area 25 centimeters squared, and height 4 centimeters.](https://cms-im.s3.amazonaws.com/ucj728GMFuFRQfH34YCinhLs?response-content-disposition=inline%3B%20filename%3D%227.6.C2.Image.13.png%22%3B%20filename%2A%3DUTF-8%27%277.6.C2.Image.13.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cf048c3b132ae0c4bcc45b080760e657e32e7db6d3d06b11a6a87c01ff0321eb)
Glossary Entries
- base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
- cross section
A cross section is the new face you see when you slice through a three-dimensional figure.
For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.
- prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
- pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
- volume
Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.