Geometry 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 area0151

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 triangle0393

(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 (y₂ - y₁)/(x₂ - x₁).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 100951

(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 number1016

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