Video Transcript
In this video, we’ll see how we can find the area of a trapezoid using two alternative formulas. We’ll also see an example of how we can apply these formulas in a real-life context. But first, let’s think about exactly what we mean by a trapezoid and the different types of trapezoid that exist.
A trapezoid is defined as a quadrilateral with one pair of parallel sides. But a word of warning, in some parts of the world, you might know this instead as a trapezium, but here we’ll use the term trapezoid. When we think of a trapezoid, we very typically think of a trapezoid that looks like this first figure or even an upside-down one, like the second figure. In fact, both of these would be called an isosceles trapezoid, and they are trapezoids in which the nonparallel sides are of equal length. But of course, by the definition, it doesn’t even need to have these nonparallel sides congruent; it just has to have one pair of parallel sides. And if a trapezoid has a right angle, then we can call it a right trapezoid.
Before we see the formula to find the area of a trapezoid, however, let’s see how we can define the different sides in a trapezoid. When we have a trapezoid, the two parallel sides are usually referred to as the bases, and often they are labeled with the letters 𝑎 and 𝑏. The two nonparallel sides are called the legs of the trapezoid. So, for example, if we were defining an isosceles trapezoid, we could say that the legs are congruent. Finally, the perpendicular distance between the two bases is the height of the trapezoid, and we usually denote that with the letter ℎ.
We will now consider how we might find the area of a trapezoid by using these letters 𝑎 and 𝑏 for the bases of the trapezoid and ℎ for the perpendicular height. We might think about splitting the trapezoid into two triangles because hopefully we recall how to find the area of a triangle. The area of a triangle is calculated as half times the base times the perpendicular height. That means then if we consider this upper triangle, the base length of this triangle is 𝑎. So, we’re working out a half times 𝑎 times ℎ, which is its perpendicular height. That simplifies to 𝑎ℎ over two. For the lower triangle, its area will be calculated as one-half times 𝑏 times ℎ, which is 𝑏ℎ over two.
Now, we know that the area of the trapezoid is going to consist of the area of the upper triangle plus the area of the lower triangle. That’s 𝑎ℎ over two plus 𝑏ℎ over two. We can then add these two fractions and take out the common factor of each, which is 𝑎 plus 𝑏 times ℎ over two. Notice that 𝑎 plus 𝑏 is the sum of the lengths of the trapezoid’s parallel bases. So, we can think of the area of a trapezoid as half the sum of the parallel bases multiplied by the height. But when it comes to finding the area of a trapezoid, don’t worry, we don’t have to always split it into triangles because in fact we have derived a general formula.
Formally, we can say that for a trapezoid of perpendicular height ℎ and bases 𝑎 and 𝑏, the area of a trapezoid is equal to one-half times 𝑎 plus 𝑏 times ℎ. Notice that this right-hand side is equivalent to calculating 𝑎 plus 𝑏 times ℎ and dividing by two. We’re really saying that the area is equal to half the sum of the lengths of the parallel bases multiplied by the height. We’ll now see how we can apply this formula to find the area of a trapezoid given its height and the lengths of its bases.
The parallel sides of a trapezoid have lengths 82 and 70. If the height is 100, what is the trapezoid’s area?
We might choose to begin a question such as this by sketching a trapezoid, recalling that a trapezoid is simply a quadrilateral with one pair of parallel sides. We are given the information that the parallel sides or bases of this trapezoid have lengths of 82 and 70. And even though these don’t have units, they would be in length units. The height, which will be the perpendicular height, is given as 100 length units. We can recall that the area of a trapezoid is given by a half times 𝑎 plus 𝑏 times ℎ, where 𝑎 and 𝑏 are the lengths of the bases and ℎ represents the perpendicular height.
For the given trapezoid, we can substitute 82 and 70 for 𝑎 and 𝑏, respectively, although it wouldn’t matter if these two values were switched. The perpendicular height ℎ is 100. Therefore, we will be calculating one-half times 82 plus 70 times 100. We can then simplify 82 plus 70, giving us 152, and half of 152 is 76. Multiplying 76 by 100, we get 7600. We weren’t given any units in the question, but as this is an area, then we can say that this will be square units or area units. So, the area of this trapezoid is 7600 square units.
In the next example, we’ll continue to use the same formula. But this time, we’ll be given the area of a trapezoid and we’ll need to work out the length of one of the parallel sides.
A trapezoid has area 1760 and the distance between the parallel sides is 40. If one parallel side is 39, what is the other side?
Let’s begin this question by visualizing the trapezoid and then filling in the information that we’re given. Firstly, we are given the area of the trapezoid. It’s 1760, and those would be in square units. Next, we are told that the distance between the parallel sides is 40 or 40 length units. Observe that we could alternatively think of this as the height or perpendicular height of the trapezoid. Finally, we are told that the length of one of the parallel sides is 39. And if we have drawn a diagram and we’re writing that on, it doesn’t matter at this point which parallel side we can label with 39.
We can recall that we can relate the two parallel sides 𝑎 and 𝑏 of a trapezoid and its perpendicular height with the formula that the area of a trapezoid is equal to one-half times 𝑎 plus 𝑏 times ℎ. Often, we use this formula to actually calculate the area of a trapezoid. But here, we are given the area of a trapezoid, so we can fill that into the formula. We can then take the value of 𝑎 to be 39, the unknown side length as 𝑏, and the height ℎ as 40. We will then have that 1760 is equal to one-half times 39 plus 𝑏 times 40.
We can begin simplifying the right-hand side by working out one-half times 40, which is 20. Then, dividing through by 20 on both sides, we note that 1760 divided by 20 is 88. So, we have 88 is equal to 39 plus 𝑏. Subtracting 39 from both sides of the equation, we have that 49 is equal to 𝑏. We can therefore give the answer that the length of the other parallel side is 49 length units.
We will now consider a different way of specifying the formula for the area of a trapezoid. Let’s start by introducing a new term, which is the middle base of a trapezoid. The middle base of a trapezoid is the line segment whose endpoints are the midpoints of the legs of the trapezoid. The middle base of a trapezoid is parallel to the trapezoid’s two bases. So, let’s consider how this is relevant to finding the area of a trapezoid. If we continue with the terminology of using 𝑎 and 𝑏 for the lengths of the bases, then the length of the middle base 𝑚 of the trapezoid is in fact the arithmetic mean of the lengths of the bases 𝑎 and 𝑏. This means that we could write that 𝑚 is equal to 𝑎 plus 𝑏 over two.
You might remember that we have seen 𝑎 plus 𝑏 over two already in this video. The concept of adding the two parallel sides and halving appears in the area formula. We can therefore replace a half 𝑎 plus 𝑏 with the letter 𝑚 in the area formula. The area of a trapezoid can be expressed as the length of the middle base multiplied by the height. Hence, we now have two slightly different but equivalent formulas. Remember that they are equivalent because one-half times 𝑎 plus 𝑏 is equal to 𝑚, which is the length of the middle base.
It just simply means that depending on the information that we are given in a particular question, whether we are given the lengths of the two parallel sides or the length of the middle base, then we can select the most appropriate formula to use. We’ll now see how we can apply this second formula in the next example.
Find the middle base length of a trapezoid that has an area of 28 square centimeters and a height of four centimeters.
Let’s consider the trapezoid in this problem. We are given the information that this trapezoid has an area of 28 square centimeters and a height of four centimeters. We need to calculate the length of the middle base of this trapezoid. We should recall that there is a formula which relates the area of a trapezoid and the length of the middle base. And it’s given as the area of a trapezoid is equal to 𝑚 times ℎ, where 𝑚 is the middle base and ℎ is the perpendicular height. We can therefore substitute in the value that 28 is the area and four is the height to give us that 28 is equal to 𝑚 times four. Then, by dividing both sides of this equation by four, we get that 𝑚 is equal to seven. The length of the middle base of this trapezoid is therefore seven centimeters.
We’ll now see one final example where we find the area of a trapezoid in a real-life problem.
A farmer owns two fields of equal area: one in the shape of a rhombus and one in the shape of a trapezoid, as shown in the figure. Calculate the length of the middle base of the trapezoidal field.
Let’s start by considering the field on the left in the figure. This is the field in the shape of a rhombus because we can see that its four side lengths are all equal. We notice that we are given the lengths of the two diagonals of this rhombus. They are 90 meters and 100 meters. We can recall that the area of a rhombus is given by 𝑑 sub one times 𝑑 sub two over two, where 𝑑 sub one and 𝑑 sub two are the lengths of its diagonals. To find the area of this rhombus then, we can take the two diagonals of 90 and 100 meters, multiply those together, and divide by two. And 9000 divided by two is 4500 square meters. So, we found the area of this field in the shape of a rhombus.
So, given that these two fields are of the same area, that means we now know that the area of the trapezoid is also 4500 square meters. But we weren’t just asked to calculate this area. We need to find the length of the middle base of this trapezoidal field. The middle base of a trapezoid is a line segment whose endpoints are the midpoints of the legs. And we should recall that there’s a formula which links the middle base of a trapezoid with its area. To find the area of a trapezoid, we multiply the length of the middle base by the perpendicular height. We can then substitute the area of 4500, and the height of the trapezoid is shown on the diagram as 25 meters.
A word of warning here just to recognize the difference between the m in 25 meters and 𝑚 of the middle base. Our equation will be 4500 is equal to 𝑚, the middle base, multiplied by 25. We can then divide both sides by 25. And without a calculator, to work out 4500 divided by 25, one method might be to divide by 100 and multiply the answer by four. This would give us that 𝑚 is equal to 180. Therefore, we can give the answer that the length of the middle base of this trapezoidal field is 180 meters.
We can now finish this video by recapping the key points. We saw that the area of a trapezoid with height ℎ and parallel bases of lengths 𝑎 and 𝑏 is given by the area of a trapezoid is equal to one-half times 𝑎 plus 𝑏 times ℎ. We can think of this informally as the area of a trapezoid is equal to half the sum of the parallel bases multiplied by the height. The middle base of a trapezoid is the line segment whose endpoints are the midpoints of the two legs of the trapezoid. The length of the middle base of a trapezoid, 𝑚, is the arithmetic mean of the lengths of the bases such that 𝑚 is equal to 𝑎 plus 𝑏 over two. Knowing this equivalence allowed us to derive the second formula that the area of a trapezoid is equal to 𝑚 times ℎ, where 𝑚 is the length of the middle base and ℎ is the perpendicular height.
