Video Transcript
In this video, we’re going to find
the area of a trapezoid using a formula, and then we’ll see how we can apply this in
real life. Let’s start by thinking about what
we mean by a trapezoid. A trapezoid is a quadrilateral,
that’s a four-sided shape, with a pair of parallel sides. So, we could draw one that looks
like this, like this, or even like this. In order to be a trapezoid, it just
has to have a pair of parallel sides. Note that in some parts of the
world, this shape is called a trapezium. And when we’re talking about
trapezoids, we may also refer to the parallel sides as the bases. We’ll now consider how we might
find the area of a trapezoid.
Let’s look at this trapezoid drawn
on squared paper. It’s got a base of four units and
another base of seven units. The height of this trapezoid would
be five units. We could, of course, count the
squares if we really wanted, but let’s see if we can find a better mathematical way
to find the area. Let’s imagine that we take a copy
of this trapezoid and rotate it so it’s alongside this one. We can see how the base of seven is
now at the top of this trapezoid and the base of four is at the bottom. The height of this trapezoid in
this position is still five. We’ve now created a parallelogram,
that is, a shape with two pairs of parallel sides.
We can use the fact that the area
of a parallelogram is calculated by multiplying the base times the perpendicular
height. So, to find the area of our
parallelogram, the base would be seven plus four, which is 11, and the height would
be five. So, 11 times five would give us the
area, and that is 55 square units. Well, that’s very nice, you might
think, but this isn’t going to get us the area of a trapezoid. But if we consider that the area of
the parallelogram is two times the area of the trapezoid, we can find the area of
the trapezoid then by taking 55 and dividing it by two, which gives us 27.5 square
units.
In this case, we had a very
specific example with specific units of four, seven, and five. So how do we get a formula for the
area of a trapezoid? Let’s say that we define our two
bases with the letters 𝑏 sub one and 𝑏 sub two, and the height we use the letter
ℎ. So, when we started to find the
area of a trapezoid, we began by finding the area of a parallelogram. So, we added our two bases, in our
case we added seven and four. And then we multiplied that by the
height of five. To find the area of one trapezoid,
we halved the area of this parallelogram. We commonly see this formula
written with the ℎ in front of the parentheses like this. So, the area of a trapezoid is
equal to a half ℎ times 𝑏 sub one plus 𝑏 sub two.
And so, that’s the formula that
we’re going to use, remembering that 𝑏 sub one and 𝑏 sub two are the bases and ℎ
is the perpendicular height. We might see this formula written
in other ways. For example, the area equals 𝑏 sub
one plus 𝑏 sub two over two times ℎ or as a half times 𝑏 sub one plus 𝑏 sub two
times ℎ. As long as we’re adding the pair of
parallel sides, multiplying by the height, and then halving it, we’ll find the area
of the trapezoid.
Just as a point of interest, there
are other ways in which we could find the area of a trapezoid. If we cut the midpoints of the
nonparallel sides, and then we can see how we could create several triangles. If we imagine chopping off this
part and sticking it into the open section here and the same on the other side, then
we would’ve created a shape which has the same area as this rectangle in green. And what’s the area of this
rectangle?
Well, the original bases were four
and seven. So, the length of this rectangle is
the same as the mean of four and seven, so we would add four and seven and divide by
two. And the height of this rectangle
would be five. This means that the area is equal
to four plus seven over two times five. That’s the same as 11 over two
times five. And working out 55 over two would
give us 27 and a half square units. Of course, here, we added the
parallel sides, divided by two, and multiplied by the height. So now, we have found two different
ways to prove how we would find the area of a trapezoid. Let’s now have a look at some
questions where we can use this formula.
The area of a trapezoid is given by
𝐴 equals a half ℎ times 𝑏 sub one plus 𝑏 sub two. Use the formula to find the area of
a trapezoid where ℎ equals six, 𝑏 sub one equals 14, and 𝑏 sub two equals
eight.
In this question, we’re already
given the general formula for the area of a trapezoid, recalling that a trapezoid is
a quadrilateral that has a pair of parallel sides. In the formula, 𝑏 sub one and 𝑏
sub two are the two parallel sides. The ℎ is the perpendicular
height. If we wanted to draw the trapezoid
that we’re given with ℎ equals six, 𝑏 sub one equals 14, and 𝑏 sub two equals
eight, it might look something like this.
But in this question, it doesn’t
really matter what it looks like because we can simply plug in the values that we’re
given into the formula. So, we’d have 𝐴 equals a half
times six, which was the height, times 14 plus eight, that’s 𝑏 sub one and 𝑏 sub
two. A half of six is three, and 14 plus
eight is 22. Three times 22 will give us 66. And that’s our answer for the area
of a trapezoid. We weren’t given any units in the
question but, of course, area units will always be square units.
Let’s have a look at another
question.
𝐴𝐵𝐶𝐷 is a trapezoid where the
lengths of its parallel bases 𝐴𝐷 and 𝐵𝐶 are 36 centimeters and 48 centimeters,
respectively. The length of the perpendicular
drawn from 𝐷 to 𝐵𝐶 is 35 centimeters. Find the area of 𝐴𝐵𝐶𝐷, giving
your answer to the nearest square centimeter.
In this question, we’re told that
𝐴𝐵𝐶𝐷 is a trapezoid. We can see this from the diagram as
we have a pair of parallel sides. So, we know it is a trapezoid. We’re told that the bases 𝐴𝐷 and
𝐵𝐶 are 36 and 48 centimeters, and that’s on the diagram. We’re also told that the
perpendicular drawn from 𝐷 to 𝐵𝐶 is 35 centimeters, and that’s also on the
diagram. Importantly, it also tells us that
the height of this trapezoid is 35 centimeters.
To find the area of 𝐴𝐵𝐶𝐷, we’re
going to use the formula to find the area of a trapezoid. This formula tells us that the area
of a trapezoid is equal to a half ℎ times 𝑏 sub one plus 𝑏 sub two, where ℎ is the
height of the trapezoid and 𝑏 sub one and 𝑏 sub two are the bases or parallel
sides of the trapezoid. So to find the area of 𝐴𝐵𝐶𝐷, we
plug in the values that we have. The height is 35 centimeters. 𝑏 sub one and be 𝑏 sub two are 36
and 48, and it doesn’t matter which way around these are. So, for the area, we’re calculating
a half times 35 times the sum of 36 and 48. We can simplify 36 plus 48 to give
us 84.
As it doesn’t matter which way we
multiply, it might seem sensible to find half of 84 rather than half of 35. This means we’re working out 35
times 42. Without a calculator, we could work
this out as 1,470. And the units here will be square
units of square centimeters. We were asked for an answer to the
nearest square centimeter, but we have an integer value here, so we don’t need to
round. Therefore, 𝐴𝐵𝐶𝐷 has an area of
1,470 square centimeters.
In the next question, we’ll find
the unknown length of the base of a trapezoid, given the other dimensions and the
area.
A trapezoid of area 132 and base 20
has height 11. What is the length of the other
base?
It might be sensible to start this
question by drawing a diagram to model the information. We’re told in this question that we
have a trapezoid. We can remember that a trapezoid is
a quadrilateral, that’s a four-sided shape, with a pair of parallel sides. The height is 11 units. We’re told that one of the bases is
20, and we need to find the length of the other base. When we talk about bases in
trapezoids, that means the lengths of the parallel sides. We don’t know which base is 20 but
let’s write it as the lower base.
In order to work out the length of
the other base, we’ll need to use the information about the area. In some countries, the word
trapezium is used to refer to a shape with one pair of parallel sides. Here, we’re told that the area is
132 square units. And we can use the formula for the
area of a trapezium or a trapezoid, which tells us that the area is equal to a half
ℎ times 𝑏 sub one plus 𝑏 sub two. ℎ represents the height of the
trapezoid, and 𝑏 sub one and 𝑏 sub two are the two bases. So, taking this formula then, we
can fill in the fact that the area is 132, the height is 11. And we don’t know one of the bases,
so let’s keep that as 𝑏 sub one. And then we add the base of 20.
We can rearrange this equation to
find 𝑏 sub one in a number of ways. But let’s start by removing this
half by multiplying both sides of the equation by two. 132 multiplied by two gives us
264. And on the right-hand side, we’ll
still have 11 times 𝑏 sub one plus 20. We can then divide both sides of
the equation by 11. 264 divided by 11 gives us 24. And on the right-hand side then,
we’ll have 𝑏 sub one plus 20. We can then find 𝑏 sub one by
subtracting 20 from both sides of the equation. And so, our answer is that the
other base must have been four units long.
We’ll now look at a real-life area
question that involves two trapezoids.
The given figure represents a
trapezoidal wooden frame. Determine the surface area of its
front side.
We can recall that a trapezoid is a
quadrilateral that has a pair of parallel sides. On the diagram, we can see that the
parallel sides would be horizontally here. We’re asked to find the surface
area or just the area of the front side. In order to find the area of this
wooden frame, we need the formula for the area of a trapezoid. This is given as the area equals a
half ℎ times 𝑏 sub one plus 𝑏 sub two, where ℎ is the height, and 𝑏 sub one and
𝑏 sub two are the bases of the trapezoid. Those are the parallel sides.
To find the area of this wooden
frame, we’re going to start by finding the area of the larger trapezoid on the
outside. This would give us the area of the
entire shape even including the center. So, we also need to find the area
of the smaller trapezoid. Then, if we subtract the smaller
trapezoid from the larger trapezoid, it’s almost like cutting out with a cookie
cutter and removing it. What we’re left with then is the
area that we want to find. Let’s begin with the large
trapezoid. We can plug the values into the
formula, but there’s a lot of information on this diagram, so we’ll be careful that
we’re using the correct ones.
The area is equal to a half times
5.5. That’s the height. And our two bases are 4.8 and six
inches. So, we’re calculating a half times
5.5 times 4.8 plus six. Simplifying the sum in parentheses,
we have a half times 5.5 times 10.8. We can then multiply a half by
10.8, which is 5.4. So, 5.5 multiplied by 5.4 gives us
29.7, and the units will be square inches. A quick reminder, if we’re
multiplying by decimals, we remove the decimal point. Here, we calculated 55 by 54. And as there were two decimal
digits in our original question, then our answer had two decimal digits. And 29.70 is the same as 29.7.
And now in the same way, we can
find the area of the smaller trapezoid. We use the same formula, and this
time the height is 4.5 inches, and the two bases are 3.4 inches and 4.8 inches. So, we’re calculating a half times
4.5 times 3.4 plus 4.8. Simplifying the parentheses then,
we’ll have a half times 4.5 times 8.2. And we can simplify by multiplying
a half by 8.2, giving us 4.5 times 4.1. So, when we’re calculating this
without a calculator, we’ll work out 45 times 41. Therefore, we know that our answer
will have the digits one eight four five, and we had two decimal digits, so our
answer is 18.45. And the units will be square
inches.
Now, we found the area of the
larger trapezoid and the smaller trapezoid. To find the area of the wooden
frame, we’re going to subtract these. So, we’ll have 29.7 subtract 18.45,
giving us the value of 11.25. And the units here will still be
square inches. And that’s our answer for the area
of the wooden frame.
We can now summarize what we learnt
in this video. Firstly, we recalled that a
trapezoid is a shape with a pair of parallel sides. We then proved that the area of a
trapezoid can be found by a half times ℎ times 𝑏 sub one plus 𝑏 sub two, where ℎ
is the height and 𝑏 sub one and 𝑏 sub two are the two parallel sides. Finally, we also noted that it
doesn’t matter which base is 𝑏 sub one or 𝑏 sub two.
