WEBVTT mathematics/basic-math/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over volume of a triangular prism.
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Volume remember is the measure of all the space inside the prism or the solid.
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Whenever you take a solid, a three-dimensional object,
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and you fill it with something, you are measuring volume, all the space inside.
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Just like for the rectangular prism, to find the volume of a triangular prism,
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you are going to find the area of the base and multiply that to the height.
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For a triangular prism, we know that the bases are triangles
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which is why it is called the triangular prism.
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When we find the area of the base, we are finding the area of the triangle.
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The height, be careful with the height because... the triangle has a height.
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But the height that we have to use here, it is not the height for the triangle.
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It is the height of the prism; it is the height between the two bases.
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Let me explain that a little bit more.
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For a volume of a prism, it is always area of the base,
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whatever the base is, times the height of the solid, of the prism.
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Since we are dealing with triangular prisms, the area of the base is going to be the area of the triangle.
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Here is the base.
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We know that the area of a triangle is 1/2 base times height.
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That is the area of a triangle.
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Be careful; like I said, this height is not the same as this height.
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The height here, this is just talking about the triangle; only the triangle.
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The height here is going to be the height of the prism.
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We know that this a base and this is a base.
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The height is going to measure the distance between the two bases, this base here to the back base.
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This right here is going to measure the height here.
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Again area of the base which is the triangle, 1/2 base times height, times the height of the prism.
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That is the volume of this triangular prism.
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Here if you look at our example, this is triangular prism.
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It might be a little bit difficult to see.
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But just think of this as the top; that is the peak.
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That is the top part; this bottom, that is the ground.
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This triangular prism, the base would be the triangles.
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Again just to review over what makes a prism,
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remember that we have two bases that are both parallel and congruent
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which will be these two triangles on opposite sides here.
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There is two sides; there is two bases always for every prism.
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They are always going to be the non-rectangle.
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If you have a prism and you have couple of different pairs that can be opposite, parallel, and congruent,
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you always look for the ones that are not rectangles because the bases are those.
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All the rest of the sides, all the lateral faces, the sides that are not bases,
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the lateral faces are going to be rectangles.
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If you look here, see that is the rectangle.
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This is a rectangle; and then the bottom side will be a rectangle.
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Again those are my bases.
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To find the volume, we are going to find the area of the base.
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For the area of the base, I can label that as capital B.
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If I do a lowercase B, then that is talking about just a base, just one side.
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That is considered the base.
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But a capital B represents the area of a base like a side.
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That is called the base; capital B is for area of the base.
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Then times the height.
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You can also think of the volume of a prism in this formula right there.
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Again triangular prism, our base is going to be the triangle.
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1/2 base times height of the triangle times the height of the prism.
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I am going to just put P here for prism.
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That is the height of the prism.
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That is just to show what type of height it is.
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For my triangle, my base and my height; remember that between these two, base and the height,
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whenever you find the area of a triangle, the base and the height have to be perpendicular.
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For this, one of these will be considered the base.
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The other one will be considered the height.
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6 and 8 will be my base and my height.
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Just remember that the base and the height, they have to be perpendicular to each other.
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1/2 times 6 times 8... I don't like the...
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You can think of it that way; 1/2 base times height.
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The height of the prism again is going to be the measure between the two bases.
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This is going to be the height of the prism; that is 10.
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1/2 times 6; 1/2 times 6; think of this as 6/1.
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Half of 6 is 3; you can also do 1/2 times 6.
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You are going to multiply whenever you do fractions.
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1 times the 6; you are going to multiply the top numbers together.
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Then multiply the bottom numbers together.
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1 times 6 is 6; over... 2 times 1 is 2; 6/2 is 3.
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3 times 8 is 24; times that by 10.
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Remember whenever you multiply a number times 10,
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you are just going to add a 0 at the end of it like that.
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Our units is centimeters.
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Remember whatever you do with volume, it is always units cubed.
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Meaning to the 3rd power; centimeters cubed, 3rd power.
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That is the volume of this triangular prism.
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The next example, again we have volume being the area of the base
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Capital B for area of the base times the height of the prism.
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The reason why I am just adding it like this is because
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this is the formula for any type of prism, not just triangular prism.
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But it could be hexagonal prism; it could be rectangular prism.
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Any type of prism, this will be the volume.
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Just whatever the base is, you are going to find the area of that base.
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Then multiply to the height of the prism.
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Here triangular prism, this is the area of the triangle.
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Let's see; 1/2 base times height; it is going to be 1/2 times the base.
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If you look at the triangle, there is a triangle.
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I am just showing you in red what the base is.
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The height of the triangle is 2; the base is 4.
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Remember the base and the height of the triangle
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when you do 1/2 base times height, they always have to be perpendicular.
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This is 2; this is 4; base times the height of 2.
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If you multiply these three numbers together, you are going to get the area of that triangle.
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Multiply it to the height of the prism, between the two bases.
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See how the front face is the base and the back is the base.
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Between them, the distance is 8; that is the height.
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Half of 4; half of 4 is 2; 2 times 2 is 4.
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The area of the base is actually 4 inches squared because that is the area.
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It is always the unit squared.
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4 times 8; volume is 32; inches to the 3rd power; units cubed.
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That is the volume of this triangular prism.
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This side right here is given to you, 5 inches.
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But that is just there to throw you off.
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Just make sure you don't use this.
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Even if they give you more measures than you need, you don't have to use it all.
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You don't need this side right here unless you need that for the area of the triangle.
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But remember the base and the height have to be perpendicular.
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That has nothing to do with the base and the height.
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That is just the length of the side right here; don't worry about that.
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1/2 base times the height times the height of the prism which is 8.
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The third example, we are going to find the volume of this solid.
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Here we have a rectangular prism and a triangular prism.
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We have to find the volume separately.
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We are going to find the volume of the rectangular prism.
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Then find the volume of the triangular prism.
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Then we are going to add them together.
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Volume of... let's do the rectangular prism first.
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For the rectangular prism, volume is remember the area of the base times the height.
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Let's say if we label the top and the bottom as the base, then we know that
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this right here is 8 meters because that is the same as that right there.
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The area of the base is 8 times this right here which is 8.
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Remember anytime you find the volume of a prism, it is going to be
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the area of the side base times the height of the prism.
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For this rectangular prism, the base is going to be 8 times the 8; 8 times 8.
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The height is this right here; 7 meters.
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8 times 8 is 64; times 7; 64 times 7.
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4 times 7 is 28; 6 times 7, 42; 44; 448.
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Volume, remember the units, it is going to be meters cubed, to the 3rd power; units cubed.
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We found the volume of the rectangular prism.
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Now I have to find the volume of this triangular prism up here.
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For the triangular prism, I am just going to put tri for triangular prism.
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The volume again same thing, area of the base times the height.
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We know that the base in this one is the triangle.
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There is our base, this triangle here.
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To find the area of that, it is going to be
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1/2 times the base of the triangle which is that right there.
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Remember the height of the triangle has to be perpendicular to that base.
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If this is the base 8, then the height is going to be 5 because that is what is perpendicular.
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See that?--it is 5.
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These three numbers multiplied together is going to give us the area of that triangle.
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Then we have to find the height of this prism.
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The height of the prism is going to be the distance between the two bases.
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If this triangle, the face in the front is the base, then the back triangle is also the base.
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The height of the prism is the distance between the two.
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How far apart are these bases?--that would be this right here.
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This is actually the height of the prism.
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We know that this side is the same as this.
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The height of that is going to be 8 or 8 meters.
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Half of 8; 1/2 times 8; we know that half of 8 is 4.
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4 times 5 is 20; remember this; that is the area of the triangle.
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Multiply that to the height of the prism.
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The volume of this is... 2 times 8 is 16.
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Then I am going to just place a 0 right there.
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It is 160 meters cubed; that is the volume of this triangular prism.
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Now I have to add them together.
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448 meters cubed plus 160 meters cubed is going to be... 8.
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This is 10; 4, 5, 6; 608 meters cubed; cubed to the 3rd power.
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This is the volume of this solid here.
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That is it for this lesson; thank you for watching Educator.com.