Back To CourseELM: CSU Math Study Guide
16 chapters | 140 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Prisms and pyramids are everywhere. If you've ever seen the cover of Pink Floyd's Dark Side of the Moon album, then you've seen a prism in action. Pyramids are even more pervasive. If you have a dollar bill in your wallet, then you're walking around with a picture of a pyramid.
Each of these shapes can be found in different forms. Let's take a closer look at what those are and then work on finding the volume of these shapes.
Let's start with prisms. You've probably seen all different kinds of prisms. There are the glass ones used for tricks with lights, like turning sunlight into rainbows. Tents can also be prisms. If you're a chocolate lover, then you know that Toblerone packages their chocolate in prism-shaped boxes.
A prism is a three-dimensional shape with flat sides and two parallel faces. What does that mean? Well, those examples above are all triangular prisms. Notice that each face is a triangle. Those are the parallel faces. And the sides? Yep, they're flat. They're also parallelograms. One way of thinking about prisms is that if you make a slice anywhere that's parallel to the face, the shape will always be the same.
Not all prisms are triangular. There are also square prisms. These include cubes where all six sides are the same, but as long as the faces are squares, it's a square prism. See, prisms are defined by those faces. If the face has five sides? It's a pentagonal prism. Also, a barn. That may be a barn.
Let's say you need to find the volume of a prism. For example, maybe you're camping and you want to fill your buddy's tent with marshmallows. You need to plan stuff like this.
Ok, the volume of a prism is pretty straightforward. Start with the area of the face. If it's a triangle, that's 1/2*b*h. If it's a square, it's s^2, and so on. Then you just multiply it by the height of the prism. So the volume of a prism is the area of the base (B) multiplied by the height between the bases (h), which we can just write as B*h.
Now, back to that tent. The front, or face, is a triangle. If the base is 4 feet long and it's 4 feet high, then the area is 1/2*4*4, or 8 square feet. Now, this tent is 7 feet long, so that's the height. The volume is just B*h, or 8*7, which is 56 cubic feet. So you're going to need 56 cubic feet of marshmallows. That's a lot of marshmallows.
Let's look at another example. Let's say you've moved up from camping pranks and you're hosting a wine and cheese party. You may have a block of cheese you're trying to slice into cubes. Each cube is one cubic centimeter. How many cubes can you get from this cheese?
This is a rectangular prism, so you need to know the area of the rectangle and the height. The face of the cheese is 3 cm long by 7 cm wide. The area of a rectangle is length times width. 3*7=21 square centimeters. The height of this block is 14 cm. So the volume is B*h, or 21*14, which is 294 cubic centimeters. So you'll have 294 tiny cubes of cheese. Oh, and if you're like me, remember to subtract a few that you'll eat while you slice.
Next, let's look at pyramids. People often confuse prisms and pyramids. But you'll never see a rainbow if you hang a pyramid in your window. When people think of pyramids, they think of the pyramids in Egypt. Why? Because they're totally awesome!
A pyramid is a shape with a base connected to an apex. The sides of a pyramid always form triangles. The bases can be any shape with three or more sides. Those ones in Egypt are square pyramids, which means their bases are squares. You can also have triangular pyramids and those with more sides to their bases.
Ok, so I bet you've always wondered, just how much stone is in one of those Egyptian pyramids? To find the volume of a pyramid, you need to multiply the area of the base (B) by the height (h), then divide by three, or 1/3*B*h. Why the 1/3? Because base times height would give you the volume of a prism. Since the top of a pyramid is a point, you know it has less volume. It's 1/3 the volume.
Ok, the largest pyramid in Egypt is the Great Pyramid of Giza. This is a square pyramid, where each side of the base is about 750 feet long. So the area of the base is 750^2, or 562,500 square feet. The pyramid originally was about 480 feet tall. So the volume is 1/3*B*h, or 1/3*562,500*480. That's 90,000,000 cubic feet.
Sounds huge, right? As a side note, the largest pyramid in the world isn't in Egypt. It's in Mexico. It's the Great Pyramid of Cholula, and its volume is over 100 million cubic feet. However, today it's covered in plants and topped with a church.
Let's look at a smaller example. Here's a triangular pyramid. It's only 6 inches tall. But hey, it's way easier to build. The base is a triangle. Remember, the area of a triangle is 1/2*b*h. It's important to not get confused with the terms. We're working with two things called base and two things called height here. But let's take it one step at a time and start with the triangle. Its base is 3 inches. And its height is 4 inches. Plug that in to 1/2*b*h and you have 1/2*3*4, or 6 square inches. That's our big B. Remember the volume formula: 1/3*B*h. That'll be 1/3*6*6, which is 12 cubic inches. Tourists may not come to see our humble pyramid, but it is a pyramid.
In summary, prisms can be triangular, square or other shapes. The volume of a prism is B*h, where B is the area of the base and h is the height of the prism.
Pyramids can have bases that are triangles, squares, or other shapes, too. The volume of a pyramid is 1/3*B*h, where B is the area of the base and h is the height of the pyramid.
When this lesson is done, you might be able to:
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Back To CourseELM: CSU Math Study Guide
16 chapters | 140 lessons