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 surface area of a prism.
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Surface area is when you are finding the area of all the sides of the solid.
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For the prism, we are going to go over rectangular prism and triangular prism.
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No matter what solid you are looking for, surface area always measures the area of all the sides.
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Here the rectangular prism, imagine if you have a box and you are trying to wrap the box.
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It is a gift for someone; you need to wrap it with wrapping paper.
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All the sides of the box, all the sides that you are covering up, is going to be the surface area.
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If we were to cut this up and unfold it, it is flat two-dimensional picture of what the box looks like.
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That is called a net; this right here is a net.
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These are all the sides of this rectangular prism.
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If we unfold it up, it is going to look like this.
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Here we have four sides right here.
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We are going to call those lateral sides, lateral faces.
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These two up here, we are going to call those the bases.
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This top and the bottom are going to be our bases.
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Surface area is going to find the area of each side.
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You are going to find the area of this side, this side, this side, all six sides.
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You are going to add them together; that is the surface area.
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For lateral area, you are finding the area of all the lateral faces only.
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Just these four and not the bases.
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Lateral area is the area of all the faces except for the bases.
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Just to demonstrate a little bit further on the net, if you have this, this is a rectangular prism.
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If you were to cut it and unfold it, you are going to get a rectangle.
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Here because it doesn't have the base, we are missing the base, this side and this side.
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This would just be the measure of the lateral faces, the lateral area.
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To find this lateral area, we know that this top part right here is the base.
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If you find the measure of each of these sides of the base, that would be like this.
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Again if we are going to cut it, let's say we are going to cut it right here.
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We unfold it; it looks like this.
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This right here would be the sides right here.
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The cut is made right here; here is the cut; here is the cut.
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To find the lateral area, we need to find the area of this rectangle.
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We know that that is going to be the length times the width.
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It is all of this times all of this.
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The area here is length times the width.
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How do we find the length and the width?
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When you are given this, this side right here, the length, we are going to call this L for length.
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It is actually going to be the perimeter of the base.
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Because remember again it is all these four sides, all of those four sides.
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Then the length is actually the perimeter of the base.
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Here we know that that is just the width, the height.
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Again to find the lateral area, you are going to find the area of this rectangle.
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But this right here, to find the length of that, it is going to be the perimeter.
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This one right here, this is side one; this is side two, side three, and side four.
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This is going to be side one, side two, side three, and side four of the base.
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It is the perimeter; that is how you would find the lateral area.
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Lateral area is perimeter of the base times the height.
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Or you can think of that as the width; the height which is this right here; the height of that solid.
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Of course to find the surface area then, we are just going to add on the two bases.
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The surface area is just going to be the whole lateral area, this whole thing.
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Then we are going to just add on the area of the base.
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Because we have two of them, we are going to just take that and multiply that by 2.
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Lateral area, perimeter of the base times the height, is going to give you this, the area of this whole rectangle.
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Surface area, because surface area is the sum of the areas of all the faces including the bases,
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we are just going to take the lateral area and then just add on the area of this base right here
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and then another area of the base which is down here; those two bases.
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Let's just do a few examples; find the surface area of the rectangular prism.
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Here again surface area, remember we are just finding the area of all the sides of this prism.
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It is like... say this is face one; this is face two.
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The front is face two; let's say this is face three.
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The back is face four; five and six.
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We have all these faces; there is five and six.
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When it comes to surface area, regardless of the bases, we are just finding the area of all the sides.
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We are just going to add them together.
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Let's say face one first.
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To find the area of face one, we are going to find the area of this top side right here.
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That is this side times this side.
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That is 10 times... what is that... 4 because that is the same as that; 10 times 4.
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The length times the width of that face is 10 times 4 which is 40 centimeters squared.
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Face two is this front... let's say that that is the front; face two.
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That is 10 times this right here which is 6.
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10 times 6 which is 60 centimeters squared for area.
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Face three is the bottom side; the bottom side, length times width is 10.
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Times the 4; that is 40 centimeters squared.
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Face four, let's say that is the back.
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That is this side times this; length times width.
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This is 10 times 6; that is 60 centimeters squared.
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If you say this is the side right here, length times width, 4 times 6, 24 centimeters squared.
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The other side, we know that is the same as this side, 4 times 6.
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We have the area of all six sides.
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There is one, two, three, four, five, six; we can just add them all together.
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40 plus 60 plus 40 plus 60 plus 24 plus 24; this is surface area.
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This will be 100; this will be another 100.
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It is 100 plus another 100 plus 48; the surface area is 248 centimeters squared.
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All I did here was just find the area of each side, all six sides, and add them together.
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That is surface area.
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Of course we can find the measure of the lateral area and then find the base.
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We can add those together; that would be a faster way to find the surface area.
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But just so you understand, surface area, it is just finding the area of all the sides and then just add them all together.
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The next one, here for this cube, we are going to actually find the lateral area and then the surface area.
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Cube, we know all the sides are congruent.
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Each side, each face is going to be a square.
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If this is 5, then this side is going to be 5.
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This side is going to be 5; all the sides are 5 inches.
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For the lateral area, remember we have to first label the bases.
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Let's say that the top and the bottom sides are going to be the bases.
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Those are my bases.
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That means to find the lateral area, I want to find the sum of
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all the lateral faces, all the faces that are not red.
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To do that, let me just demonstrate this cut again.
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Let's say I am going to cut it right there; then I unfold it.
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I am going to have one, two, three, four.
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I am going to have this long rectangle.
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This is where I cut; that is my cut.
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To find the area of this, it is just one long rectangle.
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It is just length times width.
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This right here remember is this right here.
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This is this; that is that; this is the fourth.
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To find the length of this whole thing right here, this long rectangle here,
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I just need to find the perimeter of this base.
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If this is 5, then this is 5; this is also 5, 5, and 5.
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The perimeter of this, 5, 10, 15, 20, the perimeter is 20 inches.
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That means this side right here is 20 inches.
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This height of this prism is we know 5.
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This is also going to be 5.
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To find the area of this whole thing, it is going to be 20 times 5, length times the width.
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Lateral area is 20 inches times 5 inches which is 100.
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For area, anytime you are finding area, whether it is lateral area or just area of a rectangle, inches squared.
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To find the surface area, surface area is just all the sides.
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Lateral area is just the four sides; that is not including the bases.
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But we have two extra sides; those are the bases.
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I just need to find the area of each of these bases.
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Let's do that is in red; the base is going to be 5 times 5.
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5 times 5; it is 25 inches squared.
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I have two of them; base one; the bottom one, base two.
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I just need to multiply this number by 2.
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The area of my bases is 2 times 25 which is 50 inches squared.
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All I have to do to find the surface area is...
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since I have the area of all these sides right here, the four sides,
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I just need to add on my two extra sides.
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It is 100 inches squared plus 50 inches squared.
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My surface area just becomes 150 inches squared.
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For the third example, we are going to find the surface area of a triangular prism.
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Again surface area, we are just finding the area of all the sides and adding them all together.
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We know that this front side, the triangle, and the back side,
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the one that is opposite, parallel, and congruent, are the bases.
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The other three sides are my lateral faces.
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This is a little bit different than the rectangular prism
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because the rectangular prism, we are going to have six sides.
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For this, we only have five because we have the three lateral faces which is this right here.
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That is one face; then the other side, this face right here; and then the bottom face.
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There is three; including my two bases, it will be five sides.
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But still same thing, to find the surface area, we just have to find the area of all the sides.
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The lateral area remember is still going to be the same.
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This right here, this side, it is still a rectangle.
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All of this is a rectangle.
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If I am going to cut it right here, let's say I am going to cut it right here.
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Then I am going to unfold it.
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I am still going to get... here is a rectangle here.
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There is this back side, rectangle; here is the bottom side.
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This is where I cut it, right here.
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Since this right here is 10, this is going to be 10.
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We just have to find this area of the rectangle.
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This is going to be my lateral area because this is my three sides
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that are not including my bases; not including my triangles.
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Here this side right here is this side right here; that is 5.
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This next side right here is... it is this, then this.
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That is also 5 because if you see here, they are congruent.
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They are the same; that is also 5; it goes one, two, and three.
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This right here is 6 meters.
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To find the length of this right here, this side, it is still the perimeter of the base.
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Remember the ones that we did, the rectangular prism.
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It is always going to be the perimeter of the base.
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Our base is a triangle; it is still the perimeter because we cut it here.
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It went 5, 5, and 6.
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We know this is 10; so we can find the lateral area.
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This 5 plus 5 is 10 plus 6 is 16 meters.
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To find the lateral area, I just multiply base times height or length times width.
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10 times 16; my lateral area is 160 meters squared.
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I want to find my triangle, my area of my base.
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The area of a triangle is 1/2 base times the height.
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1/2, my base is 6, my height is 4.
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Remember the base and the height of the triangle have to be perpendicular to each other.
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This is 6; this is 4.
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Half of 6 is 3; times 4 is 12; 12 meters squared.
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This is just the area for one of my bases.
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I have two of them, the front and the back.
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My bases, I have to multiply that by 2.
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Or you can just do 12 plus 12; that is 24 meters squared.
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To find my surface area, it is going to be the three sides here plus the two bases.
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The three sides together is 160.
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Plus my two bases which is this number here, 24.
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It is going to give me 184; my units squared, meters squared.
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Again surface area, you are just finding the area of all the sides.
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Then you are going to add them together.
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You can just find the area of all the lateral faces which is called the lateral area.
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Then add on the areas of your bases.
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That is going to give you surface area.
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That is it for this lesson; thank you for watching Educator.com.