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## Geometry (all content)

### Course: Geometry (all content)>Unit 7

Lesson 7: Area of trapezoids & composite figures

# Area of trapezoids

Area of a trapezoid is found with the formula, A=(a+b)/2 x h.  Learn how to use the formula to find area of trapezoids. Created by Sal Khan.

## Want to join the conversation?

• What is the formula for a trapezoid?
(26 votes)
• the formula to find the area of a trapezoid: (first base + second base) * height/2
(5 votes)
• can't you just add both of the bases to get 8 then divide 3 by 2 and get 1.5 then multiply and still get the same answer?
(15 votes)
• At what does sal mean by the average. Also this video was very helpful
(3 votes)
• That is a good question!

So what Sal means by average in this particular video is that the area of the Trapezoid should be exactly half the area of the larger rectangle (6x3) and the smaller rectangle (2x3). Think of it this way - split the larger rectangle into 3 parts as Sal has done in the video. Let's call them Area 1, Area 2 and Area 3 from left to right. Notice that:

`1. In Area 1, the triangle area part of the Trapezoid is exactly one half of Area 1`
`2. In Area 2, the rectangle area part`` of the Trapezoid is equal to Area 2 as well as the area of the smaller rectangle. Adding the 2 areas leads to double counting, so we take one half of the sum of smaller rectangle and Area 2`
`3. In Area 3, the triangle area part of the Trapezoid is exactly one half of Area 3`

Therefore, the area of the Trapezoid is equal to [(Area of larger rectangle + Area of smaller rectangle) / 2]. You can intuitively visualise Steps 1-3 or you can even derive this expression by considering each Area portion and summing up the parts. Either way, you will get the same answer.

Hope this helped.
(25 votes)
• why it has to be (6+2).3/2? not (6-2).3/2?
(0 votes)
• I'll try to explain and hope this explanation isn't too confusing!
1. Sal first of all multiplied 6 times 3 to get a rectangular area that covered not only the trapezoid (its middle plus its 2 triangles), but also included 2 extra triangles that weren't part of the trapezoid.
2. Then, in ADDITION to that area, he also multiplied 2 times 3 to get a second rectangular area that fits exactly over the middle part of the trapezoid. In other words, he created an extra area that overlays part of the 6 times 3 area.
3. So, by doing 6*3 and ADDING 2*3, Sal now had not only the area of the trapezoid (middle + 2 triangles) but also had an additional "middle + 2 triangles". That's why he then divided by 2.
I hope this is helpful to you and doesn't leave you even more confused!
(12 votes)
• hi everyone how are you today
(5 votes)
• I'm fine.:)
(2 votes)
• a rhombus as an area of 72 ft and the product of the diagonals is
144. What is the length of each diagonal?
(3 votes)
• 𝑑₁𝑑₂ = 2𝐴 is true for any rhombus with diagonals 𝑑₁, 𝑑₂ and area 𝐴, so in order to find the lengths of the diagonals we need more information.
(4 votes)
• How are you getting 12 i keep only getting 21 and doing the exact same eqation!
(4 votes)
• How do we do this because my school tablet does not let me watch the videos
(4 votes)
• how do you discover the area of different trapezoids?
(1 vote)
• By using the area of trapizoid equation, A=1/2*h*(b1+b2)
Area=1/2* height* (Base1+base2)
(6 votes)
• Okay I understand it, but I feel like it would be easier if you would just divide the trapezoid in 2 with a vertical line going in the middle.
(3 votes)
• Well, then the resulting shape would be 2 trapezoids, which wouldn't explain how the area of a trapezoid is found. But if you find this easier to understand, the stick to it. You're more likely to remember the explanation that you find easier.
(2 votes)

## Video transcript

So right here, we have a four-sided figure, or a quadrilateral, where two of the sides are parallel to each other. And so this, by definition, is a trapezoid. And what we want to do is, given the dimensions that they've given us, what is the area of this trapezoid. So let's just think through it. So what would we get if we multiplied this long base 6 times the height 3? So what do we get if we multiply 6 times 3? Well, that would be the area of a rectangle that is 6 units wide and 3 units high. So that would give us the area of a figure that looked like-- let me do it in this pink color. The area of a figure that looked like this would be 6 times 3. So it would give us this entire area right over there. Now, the trapezoid is clearly less than that, but let's just go with the thought experiment. Now, what would happen if we went with 2 times 3? Well, now we'd be finding the area of a rectangle that has a width of 2 and a height of 3. So you could imagine that being this rectangle right over here. So that is this rectangle right over here. So that's the 2 times 3 rectangle. Now, it looks like the area of the trapezoid should be in between these two numbers. Maybe it should be exactly halfway in between, because when you look at the area difference between the two rectangles-- and let me color that in. So this is the area difference on the left-hand side. And this is the area difference on the right-hand side. If we focus on the trapezoid, you see that if we start with the yellow, the smaller rectangle, it reclaims half of the area, half of the difference between the smaller rectangle and the larger one on the left-hand side. It gets exactly half of it on the left-hand side. And it gets half the difference between the smaller and the larger on the right-hand side. So it completely makes sense that the area of the trapezoid, this entire area right over here, should really just be the average. It should exactly be halfway between the areas of the smaller rectangle and the larger rectangle. So let's take the average of those two numbers. It's going to be 6 times 3 plus 2 times 3, all of that over 2. So when you think about an area of a trapezoid, you look at the two bases, the long base and the short base. Multiply each of those times the height, and then you could take the average of them. Or you could also think of it as this is the same thing as 6 plus 2. And I'm just factoring out a 3 here. 6 plus 2 times 3, and then all of that over 2, which is the same thing as-- and I'm just writing it in different ways. These are all different ways to think about it-- 6 plus 2 over 2, and then that times 3. So you could view it as the average of the smaller and larger rectangle. So you multiply each of the bases times the height and then take the average. You could view it as-- well, let's just add up the two base lengths, multiply that times the height, and then divide by 2. Or you could say, hey, let's take the average of the two base lengths and multiply that by 3. And that gives you another interesting way to think about it. If you take the average of these two lengths, 6 plus 2 over 2 is 4. So that would be a width that looks something like-- let me do this in orange. A width of 4 would look something like this. A width of 4 would look something like that, and you're multiplying that times the height. Well, that would be a rectangle like this that is exactly halfway in between the areas of the small and the large rectangle. So these are all equivalent statements. Now let's actually just calculate it. So we could do any of these. 6 times 3 is 18. This is 18 plus 6, over 2. That is 24/2, or 12. You could also do it this way. 6 plus 2 is 8, times 3 is 24, divided by 2 is 12. 6 plus 2 divided by 2 is 4, times 3 is 12. Either way, the area of this trapezoid is 12 square units.