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 area of triangles, rhombi, and trapezoids.
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First, let's go over the area of a triangle; now, we have been doing this for years now: it is 1/2 base times height.
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Now, the reason why it is half base times height: let's say I have a parallelogram.
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A parallelogram is a quadrilateral with two pairs of opposite sides parallel.
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Now, the area of a parallelogram, whether it be this type of parallelogram, a rectangle, or a square, is base times height.
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To get a triangle from a parallelogram, we have to cut it in half; if we cut a parallelogram in half, we get a triangle.
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So, we are dividing it by 2; so, the triangle is 1/2 base times height.
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Now, base times height, divided by 2, is the exact same thing.
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Think of a triangle as half the area of a parallelogram; a parallelogram is base times height, so it would just be base times height, divided by 2.
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And it is important to keep in mind that if this is the base (it doesn't matter which one you label the base,
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but it is always easiest to just label the bottom side the base), then the height has to be the length from the base
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to the vertex opposite that base, so that it is perpendicular.
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If you are going to name this the base, then this has to be the height; it is 1/2 the base times the height.
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Make sure that this is not the height; height has to be straight vertically, perpendicular to the base.
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So again, the area of a triangle is 1/2 the base times the height.
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Next is the trapezoid; now, the trapezoid formula for area is 1/2 times the height times the two bases added together, the sum of the two bases.
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Now, it looks a little long and complicated, but it is actually not; if you think about it, it is actually the same as the parallelogram.
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The area equals base times height: now, it is the same formula, but the reason why it is kind of complicated
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is because the base here...when it comes to a parallelogram, let's say a rectangle,
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we know that, if we are going to label this the base, well, this is also the base, too; this is the base, and this is the base.
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They are the same, so we don't have to worry about two different numbers for the base, because they are exactly the same.
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When it comes to a rectangle, if I talk about "base," then I could be talking about this one or this one, because they are exactly the same.
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When it comes to a trapezoid (and by the way, a trapezoid is when you have one pair of opposite sides parallel--only one),
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well, we have two different bases; and remember, bases, in this case, have to be the parallel sides.
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So, this would be one of the bases, base 1; and the side that is parallel, opposite, to it, will be base 2.
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They have to be the bases; you can't call these bases--they are the legs (these are called legs).
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But here is a base, and here is a base; now, unlike our rectangle, where these opposite sides,
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both being bases, are exactly the same--here our bases are different.
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So, for this formula, we would just have to look at this base again; it is the average of the two bases.
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We are using the same exact formula, but this represents the average of the two bases, because the bases are different.
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Now, if I rewrite this formula, I can write it as height, times base 1 plus base 2, divided by 2.
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All I did here was to take this 1/2 and put it under the two bases, the sum of the bases, right here.
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Now, if I do this, then how do I find the average?
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I have to add them up and divide by the number--whatever I have.
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So, this can be considered the average of the two bases; again, it is the same thing, base times height;
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but then, the base wouldn't just be any base, because we have two different bases; so you have to take the average of the two bases.
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So, area equals base, or the average of the two bases, times the height.
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Think of it that way; that way, it is just a little bit easier to remember the formula.
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It is the height, times the average of the bases; and that way, you don't have to think of this 1/2 in the front.
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If you want, you can just use the same formula, this formula that is written here; but you can also just use this 1/2 to make this over 2.
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And it would just be the average of the bases, times height; so it is still base times height, but it is just the average of the bases--
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the two bases, added together, divided by 2.
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And again, the height has to be perpendicular to the base.
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So, it is base times height, but the base for a trapezoid has to be the average of these two bases.
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Now, let's say I have this height being 3, and this base has a measure of 6, and this base has a measure of 8.
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Again, area equals base times height; but since I have a trapezoid, I have to find the average of the bases;
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so it is going to be 6 + 8, divided by 2, times the height, which is 3.
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6 + 8 is 14, divided by 2 is 7; so the average of 6 and 8 is 7, so 7 is actually going to be the number that we are going to use as our base.
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That, times the 3, is 21; so 21 units squared--that would be our area.
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Moving on to the rhombus: now, if you only have one, it is called a rhombus; if you have more than one, the plural is rhombi.
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Now, a rhombus is a quadrilateral (a four-sided polygon) with four congruent sides.
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Now, these angles are not perpendicular; if they were, it would be considered a square; it is just an equilateral quadrilateral.
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Now, with these four sides, they form two diagonals; there is one diagonal, and there is another diagonal:
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diagonal 1 and diagonal 2--it doesn't matter which one you call diagonal 1 and which one you call diagonal 2.
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There are two of them, and you are going to be multiplying both of them together and then dividing it by 2.
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Now, these diagonals, for any rhombus, are going to be perpendicular; so again, 1/2 times the two diagonals...
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you can think of it as diagonal 1, times diagonal 2, and then divided by 2; in this case, it is not the average of the diagonals,
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because to find the average, you would have to add up the two diagonals and then divide it by 2; here we are multiplying.
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Multiply this diagonal by this diagonal, and then divide it by 2; and that is the area of a rhombus.
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Let's go into our examples: the first one: we are going to find the area of the polygon.
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Now, since we see these little symbols right here, I know that these two sides are parallel.
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That means that, since that is the only pair of parallel sides that I have, this is a trapezoid.
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To find the area of a trapezoid, it is still going to be base times height; but because we have to different bases, we have to find the average of those bases.
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So, to find the average, we add them up and divide by however many we have.
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In this case, we have two bases, so we are going to do 9 + 11, divided by 2, times the height; and this is the height.
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Let me just do that, so that you know that that is perpendicular.
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It is going to be times 6; area equals...9 + 11 is 20; 20/2 times 6...10 times 6 is 60, and that is inches squared.
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Remember: with area, you always have to make it units squared; and that is the answer.
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The next example: Find the area of the figure.
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Now, this is a 1, 2, 3, 4, 5-sided polygon, but we don't have a formula for just any five-sided polygon.
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What you would have to do is break it up into two parts, two different polygons: we have a triangle up here, and we have a rectangle down here.
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And then, once you find the area of this and find the area of this, we just add it together.
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Let's see, for the rectangle...the area of the rectangle plus the area of the triangle...that is going to give us the area of the whole thing.
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First, the area of the rectangle: well, we know that it is base times height, so that will be base times height;
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for the triangle, remember, it is half a parallelogram; so it is just base times height divided by 2, or this.
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And we are just going to add them all up: so here, the area of a rectangle is 10 times 12, which is going to be 120.
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For the triangle, we have 1/2...what is the base?...well, it doesn't tell us what this is, but it tells us what that is;
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and we know that, since this is a rectangle, this is going to also be 12.
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So, that is 12 as the base, and the height is 8; make sure that you use the height that is perpendicular to the base.
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This is...you can, just to make it easier on you, put this over 1; and then you can cross-cancel these.
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So then, this is divided by 2, so it becomes 6; so that will be 6 times 8, which is 48; so the area of the rectangle,
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plus the area of the triangle, is going to give us 168; the units are meters squared.
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Any time you have area, you are always going to do units squared; so this is the area of this figure.
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OK, the next example: we are going to find the area of this figure.
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Let's see, we have here a rhombus; I know that that is a rhombus, because I have four congruent sides, and the diagonals are perpendicular.
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So, here is a rhombus; and this is a trapezoid, because we have one pair of parallel sides.
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I can just find the area of this, find the area of this, and then add them together.
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So first, to find the area of the rhombus: area is 1/2 diagonal 1 times diagonal 2.
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I multiply the diagonals together, and then divide it by 2: 1/2 times...
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now, if this is 4, this whole diagonal...don't just consider this; this is only half of the diagonal, so this whole thing is 8;
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and this whole diagonal...if this is 6, then this is also 6, and this whole thing is 12;
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and we are just going to multiply it all together.
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Now, you want to cross-cancel out one of these numbers; it is probably just easier to cross-cancel out the bigger numbers.
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You can just make that into a 6; 8 times 6 is 48 units squared.
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And then, for our trapezoid, area is base times the height, but remember, because we have two different bases
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(the bases are the two parallel sides), we have to take the average of those two bases, so add them up and divide by 2.
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Keep in mind: even though the bases, the two parallel sides, are here and here, and it might seem like,
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(since this is the one on the bottom--this is the side that is on the lower side, the bottom side)...that is not considered the base.
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It has to be the two parallel sides, 5 and 7; so 5 + 7, divided by 2, times the height...
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now, here they don't give us the height of this, but we can use this;
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now, this is supposed to be the same as this, so this will be 6.
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5 + 7, divided by 2...my average is 6, because this is 12, divided by 2 is 6, times 6, which is 36 units squared.
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To find the area of the whole thing, I am going to take the area of the rhombus, 48,
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and add it to the trapezoid--that is 36; and that is going to be 84 units squared.
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For the fourth example, the area of a trapezoid is 60 square inches, and its two bases are 5 and 7, and we are going to find the height.
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In this case, the area is given; the measures of the bases are given; and then, we have to find the height.
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First, let's draw a trapezoid: the area is 60...parallel, parallel; the shorter base is 5, and then 7; we want to find the height.
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h is what we are looking for: now, remember the formula for the trapezoid.
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It is the average of the bases, times the height; so it is base times height, but it is just the average of the bases.
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Here, area is 60; that equals...my two bases are 5 + 7, over 2; and h is what I am going to be looking for.
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Now, let's simplify inside the parentheses and find the average of the bases: 5 + 7 is 12, divided by 2 is 6.
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Here, to solve for h, I am going to divide the 6; so I get 10 as my height.
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And this is all in inches, so my height will be 10 inches.
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If you are given the area, and you have to look for a missing side, base, height...whatever it is,
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just plug everything into the formula and solve for the unknown variable; solve for what you are looking for.
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Make sure that you don't forget your units.
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And that is it for this lesson; thank you for watching Educator.com.