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