Topic C: Use equations and inequalities to solve geometry problems
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Let's do some solid geometry volume problems. So they tell us, shown is a triangular prism. And so there's a couple of types of three-dimensional figures that deal with triangles. This is what a triangular prism looks like, where it has a triangle on one, two faces, and they're kind of separated. They kind of have rectangles in between. The other types of triangular three-dimensional figures is you might see pyramids. This would be a rectangular pyramid, because it has a rectangular-- or it has a square base, just like that. You could also have a triangular pyramid, which it's just literally every side is a triangle. So stuff like that. But this over here is a triangular prism. I don't want to get too much into the shape classification. If the base of the triangle b is equal to 7, the height of the triangle h is equal to 3, and the length of the prism l is equal to 4, what is the total volume of the prism? So they're saying that the base is equal to 7. So this base, this right over here is equal to 7. The height of the triangle is equal to 3. So this right over here, this distance right over here, h, is equal to 3. And the length of the prism is equal to 4. So I'm assuming it's this dimension over here is equal to 4. So length is equal to 4. So in this situation, what you really just have to do is figure out the area of this triangle right over here. We could figure out the area of this triangle and then multiply it by how much you go deep, so multiply it by this length. So the volume is going to be the area of this triangle-- let me do it in pink-- the area of this triangle. We know that the area of a triangle is 1/2 times the base times the height. So this area right over here is going to be 1/2 times the base times the height. And then we're going to multiply it by our depth of this triangular prism. So we have a depth of 4. So then we're going to multiply that times the 4, times this depth. And we get-- let's see, 1/2 times 4 is 2. So these guys cancel out. You'll just have a 2. And then 2 times 3 is 6. 6 times 7 is 42. And it would be in some type of cubic units. So if these were in-- I don't know-- centimeters, it would be centimeters cubed. But they're not making us focus on the units in this problem. Let's do another one. Shown is a cube. If each side is of equal length x equals 3, what is the total volume of the cube? So each side is equal length x, which happens to equal 3. So this side is 3. This side over here, x is equal to 3. Every side, x is equal to 3. So it's actually the same exercise as the triangular prism. It's actually a little bit easier when you're dealing with the cube, where you really just want to find the area of this surface right over here. Now, this is pretty straightforward. This is just a square, or it would be the base times the height. Or essentially the same, it's just 3 times 3. So the volume is going to be the area of this surface, 3 times 3, times the depth. And so we go 3 deep, so times 3. And so we get 3 times 3 times 3, which is 27. Or you might recognize this from exponents. This is the same thing as 3 to the third power. And that's why sometimes, if you have something to the third power, they'll say you cubed it. Because, literally, to find the volume of a cube, you take the length of one side, and you multiply that number by itself three times, one for each dimension-- one for the length, the width, and-- or I guess the height, the length, and the depth, depending on how you want to define them. So it's literally just 3 times 3 times 3.