### Area of Triangles Rhombi, & Trapezoids

- Area of a triangle = ½ × base × height
- Area of a trapezoid = ½ × height × (base 1 + base 2)
- Area of a rhombus = ½ × diagonal 1 × diagonal 2

### Area of Triangles Rhombi, & Trapezoids

The area of a Rhombus = [1/2] * diagonal

_{1}* diagonal

_{2}

The area of a trapezoid = height * (base

_{1}+ base

_{2})

The area of a triangle = [1/2] * base * height.

Rhombus ABCD, AC = 3m, BD = 5m, find the area of rhombus ABCD.

- A = [1/2]*3*5

^{2}

Trapezoid ABCD, BC = 4m, AD = 7m, the height BE = 3m, find the area of trapezoid ABCD.

- A = [1/2] * height * (base
_{1}+ base_{2}) - A = [1/2]*3*(4 + 7)

^{2}

Right ∆ABC, AB = 5 in, BC = 8 in, find the area of ∆ABC

- A = [1/2]*AB*BC
- A = [1/2]*5*8

- Area of
- Area of
- Area of rectangular
- Area of rectangular
- Area of polygon ABCDE = Area of rectangular BCDE + Area of ABC
- Area of polygon

- The area of a Rhombus diagonal diagonal
- Diagonal =
- The length of the other diagonal is .

- height (base + base)
- 10 (5 + base)

- AD BC
- 10 BC

*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 Triangles Rhombi, & Trapezoids

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
- Area of a Triangle 0:06
- Area of a Triangle: Formula and Example
- Area of a Trapezoid 2:31
- Area of a Trapezoid: Formula
- Area of a Trapezoid: Example
- Area of a Rhombus 8:05
- Area of a Rhombus: Formula and Example
- Extra Example 1: Find the Area of the Polygon 9:51
- Extra Example 2: Find the Area of the Figure 11:19
- Extra Example 3: Find the Area of the Figure 14:16
- Extra Example 4: Find the Height of the Trapezoid 18:10

### Geometry Online Course

### Transcription: 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 base*0103

*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 complicated*0178

*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 bases*0975

*(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

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