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### Course: Class 8 > Unit 9

Lesson 1: Mensuration 9.1# Area of trapezoids

Area of a trapezoid is found with the formula, A=(a+b)h/2. To find the area of a trapezoid, you need to know the lengths of the two parallel sides (the "bases") and the height. Add the lengths of the two bases together, and then multiply by the height. Finally, divide by 2 to get the area of the trapezoid. Created by Sal Khan.

## Want to join the conversation?

- What is the formula for a trapezoid?(27 votes)
- the formula to find the area of a trapezoid: (first base + second base) * height/2(6 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?(16 votes)
- At2:50what does sal mean by the average. Also this video was very helpful(4 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.(28 votes)

- why it has to be (6+2).3/2? not (6-2).3/2?(1 vote)
- 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!(16 votes)

- hi everyone how are you today(7 votes)
- a rhombus as an area of 72 ft and the product of the diagonals is

144. What is the length of each diagonal?(4 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.(5 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.(5 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.(3 votes)

- How do we do this because my school tablet does not let me watch the videos(5 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)

- is this a second correct way to solve it 2*3+6*3*1/2(3 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.