Geometry Area of Triangles Rhombi, & Trapezoids

Welcome back to Educator.com.0000

For the next lesson, we are going to go over area of triangles, rhombi, and trapezoids.0002

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.0008

Now, the reason why it is half base times height: let's say I have a parallelogram.0019

A parallelogram is a quadrilateral with two pairs of opposite sides parallel.0030

Now, the area of a parallelogram, whether it be this type of parallelogram, a rectangle, or a square, is base times height.0038

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.0051

So, we are dividing it by 2; so, the triangle is 1/2 base times height.0064

Now, base times height, divided by 2, is the exact same thing.0076

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 2.0082

And it is important to keep in mind that if this is the base (it doesn't matter which one you label the base,0096

but it is always easiest to just label the bottom side the base), then the height has to be the length from the base0103

to the vertex opposite that base, so that it is perpendicular.0119

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.0127

Make sure that this is not the height; height has to be straight vertically, perpendicular to the base.0135

So again, the area of a triangle is 1/2 the base times the height.0147

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.0153

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.0169

The area equals base times height: now, it is the same formula, but the reason why it is kind of complicated0178

is because the base here...when it comes to a parallelogram, let's say a rectangle,0186

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.0200

They are the same, so we don't have to worry about two different numbers for the base, because they are exactly the same.0208

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.0217

When it comes to a trapezoid (and by the way, a trapezoid is when you have one pair of opposite sides parallel--only one),0226

well, we have two different bases; and remember, bases, in this case, have to be the parallel sides.0239

So, this would be one of the bases, base 1; and the side that is parallel, opposite, to it, will be base 2.0246

They have to be the bases; you can't call these bases--they are the legs (these are called legs).0261

But here is a base, and here is a base; now, unlike our rectangle, where these opposite sides,0268

both being bases, are exactly the same--here our bases are different.0277

So, for this formula, we would just have to look at this base again; it is the average of the two bases.0282

We are using the same exact formula, but this represents the average of the two bases, because the bases are different.0292

Now, if I rewrite this formula, I can write it as height, times base 1 plus base 2, divided by 2.0302

All I did here was to take this 1/2 and put it under the two bases, the sum of the bases, right here.0318

Now, if I do this, then how do I find the average?0327

I have to add them up and divide by the number--whatever I have.0332

So, this can be considered the average of the two bases; again, it is the same thing, base times height;0338

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.0348

So, area equals base, or the average of the two bases, times the height.0354

Think of it that way; that way, it is just a little bit easier to remember the formula.0365

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.0370

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 2.0378

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--0391

the two bases, added together, divided by 2.0398

And again, the height has to be perpendicular to the base.0401

So, it is base times height, but the base for a trapezoid has to be the average of these two bases.0409

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 8.0417

Again, area equals base times height; but since I have a trapezoid, I have to find the average of the bases;0434

so it is going to be 6 + 8, divided by 2, times the height, which is 3.0444

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.0458

That, times the 3, is 21; so 21 units squared--that would be our area.0471

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.0486

Now, a rhombus is a quadrilateral (a four-sided polygon) with four congruent sides.0504

Now, these angles are not perpendicular; if they were, it would be considered a square; it is just an equilateral quadrilateral.0511

Now, with these four sides, they form two diagonals; there is one diagonal, and there is another diagonal:0523

diagonal 1 and diagonal 2--it doesn't matter which one you call diagonal 1 and which one you call diagonal 2.0538

There are two of them, and you are going to be multiplying both of them together and then dividing it by 2.0544

Now, these diagonals, for any rhombus, are going to be perpendicular; so again, 1/2 times the two diagonals...0551

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,0568

because to find the average, you would have to add up the two diagonals and then divide it by 2; here we are multiplying.0575

Multiply this diagonal by this diagonal, and then divide it by 2; and that is the area of a rhombus.0582

Let's go into our examples: the first one: we are going to find the area of the polygon.0593

Now, since we see these little symbols right here, I know that these two sides are parallel.0597

That means that, since that is the only pair of parallel sides that I have, this is a trapezoid.0605

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.0611

So, to find the average, we add them up and divide by however many we have.0624

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.0630

Let me just do that, so that you know that that is perpendicular.0644

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.0647

Remember: with area, you always have to make it units squared; and that is the answer.0668

The next example: Find the area of the figure.0680

Now, this is a 1, 2, 3, 4, 5-sided polygon, but we don't have a formula for just any five-sided polygon.0685

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.0701

And then, once you find the area of this and find the area of this, we just add it together.0709

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.0718

First, the area of the rectangle: well, we know that it is base times height, so that will be base times height;0736

for the triangle, remember, it is half a parallelogram; so it is just base times height divided by 2, or this.0751

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 120.0763

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;0777

and we know that, since this is a rectangle, this is going to also be 12.0786

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.0797

This is...you can, just to make it easier on you, put this over 1; and then you can cross-cancel these.0807

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,0817

plus the area of the triangle, is going to give us 168; the units are meters squared.0832

Any time you have area, you are always going to do units squared; so this is the area of this figure.0847

OK, the next example: we are going to find the area of this figure.0857

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.0860

So, here is a rhombus; and this is a trapezoid, because we have one pair of parallel sides.0879

I can just find the area of this, find the area of this, and then add them together.0890

So first, to find the area of the rhombus: area is 1/2 diagonal 1 times diagonal 2.0897

I multiply the diagonals together, and then divide it by 2: 1/2 times...0917

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;0925

and this whole diagonal...if this is 6, then this is also 6, and this whole thing is 12;0941

and we are just going to multiply it all together.0950

Now, you want to cross-cancel out one of these numbers; it is probably just easier to cross-cancel out the bigger numbers.0953

You can just make that into a 6; 8 times 6 is 48 units squared.0962

And then, for our trapezoid, area is base times the height, but remember, because we have two different bases0975

(the bases are the two parallel sides), we have to take the average of those two bases, so add them up and divide by 2.0993

Keep in mind: even though the bases, the two parallel sides, are here and here, and it might seem like,1002

(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.1011

It has to be the two parallel sides, 5 and 7; so 5 + 7, divided by 2, times the height...1017

now, here they don't give us the height of this, but we can use this;1027

now, this is supposed to be the same as this, so this will be 6.1030

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.1044

To find the area of the whole thing, I am going to take the area of the rhombus, 48,1061

and add it to the trapezoid--that is 36; and that is going to be 84 units squared.1070

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.1091

In this case, the area is given; the measures of the bases are given; and then, we have to find the height.1103

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.1113

h is what we are looking for: now, remember the formula for the trapezoid.1133

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.1140

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.1149

Now, let's simplify inside the parentheses and find the average of the bases: 5 + 7 is 12, divided by 2 is 6.1167

Here, to solve for h, I am going to divide the 6; so I get 10 as my height.1186

And this is all in inches, so my height will be 10 inches.1195

If you are given the area, and you have to look for a missing side, base, height...whatever it is,1207

just plug everything into the formula and solve for the unknown variable; solve for what you are looking for.1218

Make sure that you don't forget your units.1225

And that is it for this lesson; thank you for watching Educator.com.1228