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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson and you will see that there are different kinds of prisms. You will also learn how to separate a prism into its various parts so you can easily find the surface area. You will also learn the formula for the volume.

A **prism** is a three-dimensional object with parallel straight sides and a top and bottom base that are the same polygonal shape. The base can be any shape as long as it is a polygon, meaning a two-dimensional shape with straight sides. We identify the various prisms by their bases. For example, if the base is a triangle, then we would call it a triangular prism. If the base is a square, then we would call it a square prism. Can you guess what we would call a prism with a pentagon as a base? That's right - we would call it a pentagonal prism. In real life, some skyscrapers are even the shape of prisms. Do you see how this skyscraper has parallel straight sides going up and how it has the same base at the top and bottom? This particular skyscraper has a rectangular base. Do you also see that if we took a cross-section anywhere along its length that the cross-section will always be the shape of the base?

Because the base plays a defining role in a prism, it is one of the important measurements we need in order to calculate our surface area and volume. The one measurement we need from the base part is the area it covers. We will label this measurement with a big *B*. So, remember, that when you see a big *B*, that it stands for the area of the base. If we had a square prism, then the area of our base would be the length times the width of the square. So, if our square was 4 inches by 4 inches, then our *B* would equal 16 inches squared, because the area of the base, which happens to be a square, is the length times the width of the square, so 4 * 4. If our prism had a triangular base, then we would need to calculate the area of a triangle if the problem didn't already give us the area. So, if our triangle was 4 inches across and 4 inches high, then our triangle is 4 * 4 / 2 = 8 inches squared.

The next important measurement that we need is the perimeter of the base. This we will label with a big *P*. To find the perimeter of the base, we would add up all the sides going around the base. For a square with sides that are 4 inches, our perimeter is 4 + 4 + 4 + 4 = 16 inches. If our prism is triangular with all the sides measuring 3 inches, then our perimeter is 3 + 3 + 3 = 9 inches.

The last measurement that we need is the height of the prism. So, if we had our prism standing on its base, the height is how tall the prism stands. This measurement will be labeled *h*.

Now that we have all our measurements in place, we can go ahead and use our formulas. The formula to find the surface area, the total area of just the surface, includes our area of the base, our perimeter, and our height.

Surface Area = 2 * *B* + *P* * *h*

What we are essentially doing is multiplying the area of the base by two because we have two bases and then we are adding that number to the surface area of just the sides of the prism which we find by multiplying the perimeter with the height.

To find the surface area of our square prism with a base that measures 4 inches by 4 inches and with a height of 8 inches, we first have to calculate the area of the base and the perimeter. We did this earlier, so we can use that information here. The area of the base, our square, is 16 inches squared and our perimeter is 16 inches with a height of 8 inches. We can plug all this information into our surface area formula.

Surface Area = 2 * 16 + 16 * 8

We then evaluate to find our answer. We multiply the 2 with the 16 to get 32. The 16 multiplied by 8 is 128. Adding the 128 and the 32 gives us 160 inches squared for our answer.

Surface Area = 2 * 16 + 16 * 8

Surface Area = 32 + 128

Surface Area = 160 inches squared

Our answer ends with our measuring units squared because area is always squared.

The formula for volume, the amount of space inside the three-dimensional object, is simpler than the one for surface area. It only requires the area of the base and the height.

Volume = *B* * *h*

All we have to do is find the area of the base and multiply it by the height to find the volume.

For our square prism, we already know that the area of our base is 16 inches squared. Our height is 8 inches, so we can plug that information into our formula.

Volume = 16 * 8

To find our answer, we multiply the 16 by the 8 to get 128 inches cubed for our answer.

Volume = 16 * 8

Volume = 128 inches cubed

Because volume is always cubed, our answer ends with our measuring units cubed.

What have we learned? We've learned that **prisms** are three-dimensional objects with parallel straight sides and top and bottom bases that are the same polygonal shape. Some skyscrapers are shaped like prisms. Prisms are identified by the shape of their base. A prism with a triangle base is called a triangular prism. A prism with a square base is called a square prism. And a prism with a pentagon base is called a pentagonal prism. The formula to find the surface area of a prism is Surface Area = 2 * *B* + *P* * *h*, and the formula to find the volume is Volume = *B* * *h*. The *B* stands for the area of the base, and the *P* stands for the perimeter of the base. The *h* is for the height of the prism.

After this lesson is done, you should be able to:

- Name prisms of all types
- Identify the measurements needed to find base, perimeter and surface area
- Demonstrate how to measure volume

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Geometry: High School15 chapters | 160 lessons

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