WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over the surface area of prisms and cylinders.
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First, let's talk about prisms: prisms, we know, are solids with two bases that are both parallel and congruent.
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They are congruent polygons that lie in congruent planes; they have to be parallel and congruent.
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If you look at this prism here, this top and this bottom right here would be the bases; they are both parallel and congruent.
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Now, the other sides, the remaining faces, are lateral faces; they are all of the faces that are not the bases.
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And each of them, each of those lateral faces, would be a parallelogram.
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So, if you notice, this is a parallelogram, parallelogram, parallelogram; all of them will be parallelograms.
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Lateral edges are those line segments where the faces intersect; so this face and this face intersect here--those are lateral edges.
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And the lateral edges would be only from all of the lateral sides, where the lateral sides intersect--not these.
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These are considered edges, too, but they are not lateral edges, because these are edges from the base.
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It would be all of these here; those are lateral edges.
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And the altitude measures the height of the prism; so altitude, we know, has to be perpendicular to the segments.
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This perpendicular segment is the altitude; it measures the height, because height, we know, has to be perpendicular.
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The two types of prisms: it is going to be either right or oblique.
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A **right prism** is when these lateral edges are altitudes; so the measure of the lateral edge, the length, is the height of the prism.
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So, this right here measures the height; then it is considered a right prism, because it is standing up (right).
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An **oblique prism** is one that is tilted; it is slanted to the side.
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So, in this case, these lateral edges are not perpendicular; they are not considered altitudes, so they are oblique; that is a prism that is not right.
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So then, if you have to find the height of this, then you would have to find the height that is perpendicular to the base.
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Classifying prisms: classify a prism by the shape of its bases.
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Depending on the bases, there are different names for these prisms.
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These are all prisms; all four of these are just a few types of prisms.
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The first one: if we label the top and the bottom as bases...
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now, for this rectangular prism, it is a special type of prism, because we can actually name any pair of opposite sides as its bases,
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because we know that bases just have to be congruent and parallel;
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and for this, each pair of opposite sides is congruent and parallel.
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So, for a rectangular prism, it doesn't matter which two opposite sides you label as bases.
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But if it is standing this way, then just to make it easier, you can name the top and the bottom as bases.
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Then, this would be a right (because it is standing upright; all of the lateral edges are perpendicular) rectangular prism.
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This one here...now, be careful here, because in this case, the bottom is not considered the base.
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This triangle right here, with this front side and this back side, would be considered the bases.
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The other sides, the bottom, the right side, and the left side, are all lateral faces.
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So, don't always think that the top and the bottom are going to be the bases; in this case, it is the front and the back, and they are triangles.
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The bases are triangles; that would make this...
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and for determining whether it is right or oblique, if we were to take this solid and stand it up,
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so that the bases were the top and the bottom sides, then this would be the height; it would be the altitude.
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And it would be perpendicular to the bases; so this is also a right...
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and then, the shape of the base is a triangle; so this is a triangular prism.
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This one here, we know, is oblique, because we can just tell that it is not standing up straight; it is slanting to the side, so this is oblique.
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And then, the bases would be this top and the bottom; and they are 1, 2, 3, 4, 5...5 sides, so that is a pentagon: pentagonal...this is "al"...prism.
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And this last one here, we know, is right; and how many sides is the base?
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We know that this top and the bottom is a base, again; 1, 2, 3, 4, 5, 6...so that is a hexagon, so that is a hexagonal prism.
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OK, to find the lateral area of a prism, first make sure that you determine which sides your bases are and what your lateral faces are.
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And once you do that, your lateral area is just the area of the lateral faces.
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If we say that the top and the bottom are the bases, that means that we are finding the area of all of these four sides.
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Left, front, right, and back--we are looking at the area of all the four sides, minus the bases--not including the bases--and that is lateral area.
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The formula to find the lateral area of a right prism would be the perimeter of the base, times the height.
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And the reason for this formula is (I am going to explain it to you): let's say that we take scissors, and we cut one of these sides.
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Let's say you cut it there--cut that corner--and you unfold it.
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When you unfold it (and I have a paper here to demonstrate), here is the rectangular prism.
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We know that it is a rectangular prism, because the bases are a rectangle.
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If you were to cut it, and you unfold, you get a rectangle; so again, this is a rectangular prism;
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if you cut it and unfold, then you get a rectangle.
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This is my cut that I made; you end up getting just a big rectangle.
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So, if this is side 1, this right side; this front side is side 2; this is side 3 on the left; and the back is side 4;
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well, it is as if I have side 1, side 2, side 3, and side 4; and actually, this cut is made to this side, the left side, or the right side of side 1.
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So, it is as if this would be side 1; so either way, it is just a big rectangle.
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So, each of these sides, 1, 2, 3, 4, 1, 2, 3, 4...then, when you fold it back up, it will be like this, with side 2 folding this way, this way, and this way.
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It is as if you are taking this, and you are folding it back over.
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Now, in that case, since the lateral faces all make up the big rectangle (I am going to erase this, so you don't get confused),
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let's say that this right here has a measure of 2, and let's say that each of these is the same;
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well, then I know that this is 2; this is 2; 2; and 2; and the height, the altitude, is, let's say, 10.
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Then, I know that this right here would be 10.
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The perimeter here: to find the area of this, it would be all of this length right here, times this.
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Here, this is 2, 4, 6, 8; this has a measure of 8, and this is 10; so the area of all of these lateral faces would be 80.
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So then, that is how this formula came about, because, if you were to cut it, well, then the area of all of the lateral sides
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would be the perimeter of the base (because you are doing this, this, this, and this) times the height,
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because if you were to unfold it, this, that, and all of these sides would come up to 8.
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And isn't that the perimeter of the base?
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So, that is why it is the perimeter of the base, which is this, times the height; and that is 10.
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So then, the lateral area of this would be 80 units squared, because it is still area; you are finding all of the space.
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That is 80 units squared, and that would be only the lateral area.
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Now, next, let's go over surface area: the surface area is the area of all of the sides, so it would be the lateral area, plus the area of the two bases.
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It is the lateral area, plus 2 times the area of the base.
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This top right here is the base, and then this right here is also a base.
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Now, if you want, you could find the area of each one of these: 1, 2...all of the sides, all of the bases, and then just add them all up.
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That would give you the surface area, because it is the sum of the areas of its outer surfaces--all of the sides.
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But since we know that the lateral area would be the perimeter--so again, if you were to cut it,
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let's say, right down here, and unfold the lateral area, then it would just be the perimeter,
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because it would be this side, this side, this side...all of those sides, which is the perimeter, times the height.
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It is going to just give you one big rectangle; the perimeter of the base is going to be the length; so this is the perimeter of the base,
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and this is the height; so that, plus...and then the area of the base, times 2, because you have 2 bases.
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And then, that would give you the area of all of the sides together.
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That is surface area: lateral area is just the area of all of the lateral sides, and then the surface area would be
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the area of all of the outer sides, including the bases; that is lateral area, plus the area of the bases.
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Next, we have cylinder: a cylinder has two bases that are parallel and congruent circles.
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So, it is like the prism, except that the bases are circles instead.
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Here is the base here, and here.
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The **axis** is the segment whose endpoints are the centers of the circles.
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So, it has to go from the center of one base to the center of the other base; that is the axis.
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Now, it could be different from altitude; now, in this case, in a right cylinder, the axis is the same as the altitude, because the altitude measures the height.
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If it is standing up straight, then it doesn't matter where the endpoints are--center to center or from end to end.
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As long as the endpoints are on the two circles, that is the altitude.
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For a right cylinder, altitude is the same thing as the axis; in an oblique cylinder, that is not the case.
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The altitude is right here; that measures the height; this is the altitude.
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But the axis, remember, has to go from the center to the center of the two circles; so this right here is the axis.
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It is not the same in an oblique cylinder.
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Now, the lateral area of a cylinder is the same concept as the lateral area of a prism; lateral area is the same.
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To find it within the cylinder, again, it is just the area of everything but the bases.
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So, we know that this is the base; we know that this is the base; so it would be the area of just the outer part.
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Now, think of a can, like a soup can; and you tear off the label--the label goes around the can.
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It is like finding the area of that label; that would be like lateral area.
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Again, if you make a cut, like this paper here--if you have a cut--there is your cylinder without any bases,
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and you cut it, and then you open it up, you are going to get a rectangle.
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To find the area of the rectangle (the lateral area just means that you are finding the area of a rectangle),
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you need base, and you need height; the base would be (if I turn it back into a cylinder)...
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isn't the base...let me actually call it the width--the width and the length--so that you don't get it confused with these bases.
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The width of the rectangle is the same as the measure of this other circle.
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What is that called? That is called circumference: to measure this all right here, that is the circumference.
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The width of this rectangle is 2πr, because it is the circumference--just 2πr.
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And then, the length, we know, is h; it is the height of that.
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That is how we get this formula here: 2πr times the height--that is the lateral area.
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And for surface area of a cylinder, the same thing works: lateral area, plus the area of the two bases.
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Now, this is the easiest way to remember it, because the lateral area is always just going to be a rectangle.
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And then, we just find the area of the circle, and the area of the other circle--
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or find one of them, and then, since they are going to be the same, just multiply it by 2,
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and then add them together: so it is this area, times 2, plus that; all of these together is going to equal the surface area of the cylinder.
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It is going to be πr², so it is 2 times πr².
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Let's do some examples: Find the lateral area and surface area of the prism.
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Now, the first thing to do is to figure out what the bases are.
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Whenever you have a prism, the easiest way to point out the bases is to look for any shapes, any sides or bases, that are not rectangular.
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Automatically, we know that these two triangles will be the bases.
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Now, if you have a face that is not rectangular, but there is only one of them, then it can't be a prism; it is going to be something else.
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It will probably be a pyramid or...I don't know; it is not going to be a prism.
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If it has two opposite sides that are congruent and that are not rectangular, then those two sides would be the bases of the prism.
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Those are my bases; and then, the lateral area--imagine the cut; you are going to unfold; it is going to be one rectangle.
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And that is three sides, three lateral faces, that make up the rectangle, this lateral area; the three sides go like that.
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One side is 6--that is one side; the other side is 6, because this side and this side are the same; and then, the other side will be 5.
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So, if this is where the cut is made, then it is as if this is where the cut is made; so 6 + 6 is 12, plus 5 is 17;
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so this whole thing right here is 17, and then the height is 8.
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We are going to do 17 times 8 to find the lateral area: on your calculator, do 17 times 8, which is going to be 136 units squared.
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That is lateral area; then, surface area--all I have to do for surface area, since I have my lateral area,
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is to find the area of the base, multiply it by 2 (since I have 2 of them), and then add it to this lateral area.
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Find the area of the triangle: now, to find the area of this triangle, it is going to be 1/2 base times height; this is 6; this is 6; and this is 5.
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I need the height; now, for the height, because this is an isosceles triangle, I know that this is half of this whole thing,
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so this will be 2.5, half of 5; then, to find the height, to find h (this is 2.5, and this is 6), I can just use the Pythagorean theorem.
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So, h² + 2.5² = 6²; 2.5 squared...h squared is 29.75;
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take the square root of that, and my height is 5.45.
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1/2...my base is 5, and then, for this base, make sure that it is the whole thing; it is not just this half.
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We only use that half just to look for the height; we can use the Pythagorean theorem for the height and make this a right triangle.
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But when it comes to the actual triangle, we are finding the area of the whole thing, so you have to use 5 as the base.
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And the height is 5.45; I am just using my calculator that I have here on my screen; I get that my area of this triangle is 13.64.
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My surface area is going to be 136 + 2 times the area of the triangle, and I got 13.64.
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Multiply it by 2, and add it to 136; and I get 163.27 units squared.
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That is the lateral area and surface area of this prism here.
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The next example: Find the lateral area of the prism.
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Now, this doesn't look like a prism; it looks kind of odd-shaped.
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But remember: as long as you have two sides that are opposite and congruent, it doesn't matter what shape it is; those are the bases of the prism.
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In this case, for this solid, we have the front and the back as the bases; this whole thing right here is considered the base.
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So, that way, each of the lateral faces is rectangular; they are all rectangles.
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Again, if you were to take this solid and stand it up, so that the bases would be the top and the bottom,
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and then make a cut like this, it is going to be the perimeter...if you unfold it, it is one long rectangle;
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and then, this right here, the length of the lateral area, is going to be the perimeter of the base.
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So, it is going to be like this--all of this is going to make up this whole thing right here.
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There is our cut; then this is like having a 1; and this is 5; that is this and then this, and then that would be the 4; and so on, all the way through.
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I am going to do perimeter as 1 + 5 + the 4 + 8 +...and then, what is this side here?
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This side would be the 4 plus the 1; this is 5, plus...and then, this whole thing right here is the 8, plus the 5; that would make this...and that is 13.
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The perimeter I get: this is 6, plus 4 is 10, plus 8 is 18, plus 5 is 23, plus 13 is 36.
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So, this is 6; this is 10; this is 18; this is 23; and then, together, they are 36.
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And so, after that, we need to find the height; this whole thing right here is 36, and then what is this height right here?
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That is 2, because all of these have to be the same; so my lateral area is going to be 36, my length, times the width, and that is 2;
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so, that will be 72 units--whether it is inches, feet, and so on--squared.
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And then, let's find the surface area of that same figure.
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The lateral area was 72 units squared; then I want to find the area of my base, because, remember:
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surface area would be the sum of all of the sides, so it will be lateral area, plus the area of the base, plus the area of the other base.
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So, it is 2 bases; we have to add both to the lateral area to get surface area.
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Here, to find the area of this base right here (just this front--this is a base), I need to break this up,
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because there is no way that I can find the area of that, unless I break it up into 2 polygons, like that.
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Here, this will be 8 times...what is that?..this is 4, and then this is 1, so then this would be 5.
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So again, length times width here--this is 40 units squared, and then, for this right here, it would be 5 times 1; so that is 5 units squared.
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Then, I add these together, and this would be 45; so the area of the base is 45 units squared.
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But then, since I have two of them--I have a front, and I have a back--my surface area is going to be my lateral area,
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all of that, plus two times my base; so that is 45; so 72 +...2(45) is 90...
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and then this will be 162 units squared; this is my surface area, then.
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The fourth example: Find the lateral area and the surface area of the cylinder.
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Now, in the same way as our prism, if we make a cut right here, and lay it out flat, then it will just be a rectangle,
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whereas this is the circumference, because this measures from here all the way around here, and that is the circumference.
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So, it is 2πr, and then this is the height, which is 9.
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2πr is 2 times π times...the radius is...4; that is the length; the width is 9.
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Multiply that together; 2 times 4 times 9...you get 226.19...now, I am just rounding to the nearest hundredth, 2 places after the decimal.
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That would be the lateral area; now, you can probably just leave it in terms of π, if you can.
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2 times 4 times 9: 2 times 4 is 8, times 9 is 72; so you can probably leave it as 72π units squared.
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But otherwise, if you have to solve it out, then you can just use your calculator: 72 times π, which is 3.14.
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This would be the answer for the lateral area.
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And then, to find surface area, we are going to find the area of the base, which is a circle.
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The area of a circle is πr²; π times r, the radius, is 4 squared, which is π, or 3.14, times 16.
00:37:17.200 --> 00:37:35.000
16 times π is 50.27 units squared, and that is the area of one of the circles.
00:37:35.000 --> 00:37:52.300
But since I have two of them, I need to multiply this by 2; so my surface area is my lateral area, plus 2 times the area of the base.
00:37:52.300 --> 00:37:57.100
And I am going to put a capital B there, to represent the area of the base:
00:37:57.100 --> 00:38:12.600
this is 226 (I am going to use this number up here) and 19 hundredths, plus 2 times 50.27.
00:38:12.600 --> 00:38:47.000
And then, using your calculator, solve that out; and you should get 326.72 in...don't forget...units squared; that is the surface area.
00:38:47.000 --> 00:38:50.000
OK, well, that is it for this lesson; thank you for watching Educator.com.