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### Area of Parallelograms

- Parallelogram: Polygon with two pairs of parallel sides
- Area = base × height
- The area of a region is the sum of the areas of all of its non-overlapping parts

### Area of Parallelograms

A parallelogram always have two pairs of parallel segments.

The area of a parallelogram equals to base times height.

Parallelogram ABCD, ―AE ⊥―BC , BC = 12, AE = 5, find the area of the parallelogram.

- Area = BC*AE
- Area = 12*5 = 60

―AE ⊥―BC , AB = 18, BC = 20, m∠BAE = 30

^{o}, find the area of parallelogram ABCD.

- AE = ABcos30
- AE = 18*0.866
- AE = 15.6
- Area = BC*AE
- Area = 20*15.6 = 311.8

The area of ∆ABC is 40, find the area of the parallelogram.

- 40*2 = 80

The area of a parallelogram is the same of the rectangular with same base and height.

Parallelogram ABCD, ―AE ⊥―BC ,AB = 10, BC = 15, m∠BAE = 30

^{o}, find the area of trapezoid AECD.

- AE = ABcos30
- AE = 10*0.866
- AE = 8.66
- Area of parallelogram ABCD = BC*AE
- Area of parallelogram ABCD = 15*8.66 = 129.9
- BE = ABsin30
- BE = 10*0.5 = 5
- Area of ∆ABE = [1/2]BE*AE
- Area of ∆ABE = [1/2]*5*8.66 = 21.7
- Area of trapezoid AECD = 129.9 − 21.7 = 108.2

Find the area of polygon ABCDEF.

- Area of polygon ABCDEF = Area of ABCG - Area of FEDG
- Area of ABCG = AB*AG = 16*(12 + 15) = 432
- Area of FEDG = DG*ED = (16 − 10)*15 = 90
- Area of polygon ABCDEF = 432 − 90 = 342.

A parallelogram always has two pairs of congruent segments.

A rectangle is also a parallelogram.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Area of Parallelograms

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Parallelograms 0:06
- Definition and Area Formula
- Area of Figure 2:00
- Area of Figure
- Extra Example 1:Find the Area of the Shaded Area 3:14
- Extra Example 2: Find the Height and Area of the Parallelogram 6:00
- Extra Example 3: Find the Area of the Parallelogram Given Coordinates and Vertices 10:11
- Extra Example 4: Find the Area of the Figure 14:31

### Geometry Online Course

### Transcription: Area of Parallelograms

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over area of parallelograms.*0002

*Now, a parallelogram, remember, is a polygon with two pairs of parallel sides.*0007

*So, these two sides are parallel, and these two sides are parallel.*0013

*To find the area, we are going to do the base times the height; so it is the same formula as a rectangle or a square.*0019

*But just remember that, although this is the base, the height has to be the length that is perpendicular to the base.*0029

*So, it doesn't actually matter which side you label as the base; this can be the base; this can be the base; this can be the base.*0040

*It does not matter, as long as the height is the length perpendicular to that base.*0051

*If I am going to call this the base, then the height has to be from here to here, so that it is perpendicular.*0061

*This would be the height; be careful not to make this the height.*0077

*If this is the base, then this is not the height; if you were to measure how tall you were, your height, you would stand up straight.*0084

*You would stand perpendicular to the ground to measure your height, just like this.*0094

*You are not going to stand to the side; you are not going to bend over and then measure you height from there.*0101

*You have to stand up straight--that is perpendicular to the ground.*0109

*That is the area: it is the base times the height; and again, the height has to be perpendicular.*0113

*To find an area of a figure that is not a parallelogram or a rectangle or any of the types of polygons that you are used to,*0123

*then you are going to have to break it up into parts.*0134

*So here, these lines are drawn for you; but if you were to just have this figure here, and it said "find the area,"*0137

*then you can break it up into parts: you can break it up into here and here; then you would have to find the area*0151

*of this rectangle here, find the area of this rectangle here,*0157

*and then find the area of this triangle here, and then we are going to add them all up--it is this plus this plus this.*0161

*Or you can maybe cut it down here; this would be one big rectangle; and then, this can be a trapezoid.*0168

*Now, we are going to go over trapezoids in the next lesson; but if you know the formula to the trapezoid,*0178

*you can add up this rectangle and the area of that trapezoid together, too.*0184

*That would be the area of figures; now, let's go over our examples.*0192

*Find the area of the shaded area, or shaded region: that would be everything in blue.*0197

*That means that you would have to find the area of this whole thing (that is the area of this parallelogram here), and subtract the area of this rectangle.*0206

*Now, I am going to tell you that is not good to assume that this is a parallelogram and this is a rectangle.*0220

*But I am just going to tell you that it is: this is a parallelogram, so show that it is a parallelogram, like this: this side with this side.*0229

*And then, I am going to draw that to show that it is a rectangle.*0240

*Again, take the area of the parallelogram, and then subtract the area of the rectangle.*0248

*It is as if I have a full parallelogram, and then you are going to cut out this rectangle; don't you have to subtract it and take it away?*0255

*The area of the parallelogram is going to be base times height; the base is 12;*0266

*the height is not 10--it has to be the perpendicular one, so it is going to be 9.*0281

*12 times 9...that is going to be 108; and then, you are going to subtract the rectangle.*0287

*The area of a rectangle is going to be base times height here; so that is 8 times 5; that is going to be 40.*0302

*And then again, you are going to take the area of the parallelogram, minus the area of the rectangle, and that is going to be the area of the shaded region.*0319

*So then, it is going to be 108...subtract the 40...and that is going to give you 68; that would be units squared,*0332

*because any time you are looking at area, it is always units squared.*0348

*The next example: Find the height and the area of the parallelogram.*0362

*We have to first look for the height; they give us the base; they give us this length; and then they give us this angle measure.*0368

*Well, if this is 45 degrees, this is a right triangle; using this right triangle, I can find this measure right here.*0379

*Here (remember special right triangles?), if you have a 45-45-90 degree triangle*0393

*(if this is 45, this has to be 45, so this is a special right triangle: 45-45-90), if this is n, the measure of the side opposite the 45-degree angle--*0402

*so then, if this is 45, then the side opposite this angle is going to be this side right here;*0416

*and the side opposite this angle is that side; so this angle and this side go together.*0423

*This is 45-degree; this is n, with that side opposite; then the side opposite this 45 is also going to be n, because they are the same; it is isosceles.*0432

*Then, the side opposite the 90, which is the hypotenuse, is going to be n times √2.*0444

*Now, that is this side right here, which is 10; so they give you this side--this is 10.*0453

*That means that n√2 is equal to 10; so, I can just make n√2 equal to 10.*0465

*Do you remember this section from special right triangles?*0482

*A 45-45-90 degree triangle is going to be n, n, n√2; so I am just going to make these two equal to each other: n√2 is equal to 10.*0486

*Then, I am going to solve for n, because this is what I want to solve for.*0498

*I want this measure here, because that is going to be the height.*0502

*Even though this is n for now, this is also h for height.*0507

*So, if I solve for n here, I need to divide the √2; so n = 10/√2.*0511

*Remember: I have to rationalize this denominator, so I have to multiply this by √2/√2.*0521

*This becomes 10√2/2; if I divide this, I am going to get 5√2, so n is equal to 5√2; this is 5√2.*0528

*Since that is also the height, I can just go ahead and say that my height is 5√2.*0545

*So, I have my base, and I have my height; now I can solve for the area.*0556

*Area equals 12 times 5√2; this is going to be 60√2 units squared.*0561

*Or, if you want, you can just punch it in your calculator; I have my calculator here, and that will be 84.85 units squared.*0580

*Example 3: Find the area of the parallelogram, given the coordinates of the vertices.*0612

*For coordinates, since these are the sides of the parallelogram, we want to know what we can name as the base--*0621

*what we can measure as the base, the length, the distance between the points--as the base and the height.*0634

*Remember: the base and the height have to be perpendicular to each other.*0641

*So, we have to make sure that the distance between these points is going to be perpendicular,*0643

*which means that their slopes have to be negative reciprocals.*0649

*If you are a visual person, you can go ahead and graph these points out, just to help you visualize what the parallelogram is going to look like.*0657

*If I have (10,-4), that is going to be in quadrant 4; so, this is 10; this is -4.*0673

*And then, (4,-4)...let's say this is (4,-4)...is right there; (4,-2)...this is (4,-2)...is right there; and (10,-2) is right there.*0687

*We know that this is going to be a rectangle; how do I know?--because these two points have the same y-coordinate;*0705

*these two points have the same x-coordinate; and the same here--these have the same x-coordinate; this has the same y-coordinate.*0715

*So, these two are horizontal, and these two are vertical.*0722

*Remember that rectangles are a special type of parallelograms, because with rectangles, the opposite sides are parallel.*0730

*So then, that is also a parallelogram.*0739

*Now, we know that, in a rectangle, these sides are perpendicular to each other; so we don't even have to worry about finding slope.*0743

*If you need to, then you would have to find slope; if you remember, the slope, m, is (y _{2} - y_{1})/(x_{2} - x_{1}).*0753

*We don't have to do it for this one; but if you have a problem like this,*0766

*where you have to find slope to see if they are actually perpendicular,*0770

*then just take the y-coordinate, subtract it from the other y-coordinate, and divide it by the difference of the two x-coordinates.*0774

*That is going to give you slope; and then, you would have to do that for all of the lines to see if they are negative reciprocals of each other,*0785

*meaning that, if the slope of two points was 2, then the negative reciprocal is going to be -1/2.*0793

*It is the negative, and then the reciprocal; so if this is 2/1, then it has to be 1/2,*0807

*and then that would mean that this slope and this slope are perpendicular to each other.*0812

*Here, since we know that they are perpendicular, we can just call this the base and call this the height.*0823

*To find the area, I know that this base is going to be 6, because it is 6 units away, from 4 to 10; the base is 6;*0834

*and then, the height...this is -2, so the measure from here to here is 2; so the area is going to be 12 units squared.*0846

*And the fourth example: we are going to find the area of this figure.*0872

*We need to break this up into several parts; you can do this different ways.*0878

*You can cut it here; you can cut it here if you want; you can cut it here, and then cut it here, if you like.*0886

*It doesn't matter; let's see, what do I want to do? Let's just do it this way.*0892

*I am going to cut it here (so then, we have this rectangle), and then cut it here (so we have this rectangle).*0900

*Now, be careful; since I cut this up, I can't use this for the base; I have to use this as the base.*0907

*So, here, this area of this is going to be 7 times 10 (the base times the height--make sure that it is perpendicular;*0916

*if you have a parallelogram, make sure you have the height); so this would be 70.*0927

*Then, this: do I know what the base is here?*0934

*Well, if this whole thing is 10, and this one is 7, then this has to be 3.*0939

*And let's see, I don't know what this is; well, if this is 3, this is 3, and from here all the way to here is 10*0951

*(because that is what this number tells me), this is 3, this is 3, and that is 6; that means that this has to be 4,*0966

*because this plus this plus all the way down here has to be 10, so this is 4.*0975

*That means that, for this rectangle here, I am going to make 3 the base and 7 the height.*0983

*Make sure that you don't multiply this; this thing right here is going to be 3 + 4, so this is going to be 3 times 7, which is 21.*0993

*So, let me just circle it, so that you know, and you don't get confused with my numbers.*1003

*And then, for this rectangle here, we have 11, and then we know that this is also 4, because we found it there.*1008

*It is just going to be...from here to here (make sure that you don't use this number, because this number*1016

*is showing you from here all the way to here) is going to be 11 times 4, which is 44.*1019

*So now, all we have to do is add up these numbers: so 70 + 44 + 21...we have three rectangles,*1029

*so we are adding up 3 numbers; so 4 + 1 is 5; 7 + 4 is 11, plus 2 is 13;*1041

*that means the area of this figure here is going to be 135 units squared.*1050

*And that is it for this lesson; thank you for watching Educator.com.*1063

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