## How Many Trapezoids I Can Draw

I guess this is a good time to give my answer for the challenge of how many different trapezoids there are to draw. At the least it’ll provide an answer to people who seek on Google the answer to how many trapezoids there are to draw. In principle there’s an infinite number that can be drawn, of course, but I wanted to cut down the ways that seem to multiply cases without really being different shapes. For example, rotating a trapezoid doesn’t make it new, and just stretching it out longer in one direction or another shouldn’t. And just enlarging or shrinking the whole thing doesn’t change it. So given that, how many kinds of trapezoids do I see?

I make it out to be six.

Here’s the way I reasoned it. For simplicity, I’m assuming the two parallel bases are horizontal. And I’m assuming the lower base is the longer one; this is the way trapezoids keep getting drawn. I may as well go with the universal standard. I’m also assuming that just making both bases or both legs longer doesn’t by itself change the trapezoid. But I do think there are some differences, and this is how I come up with six.

The difference I was thinking of came about from a comment Chiaroscuro made about how to prove the area formula for a trapezoid. His idea, and a good one, was based on slicing the trapezoid up into three shapes. One of them would be the triangle on the left-hand side, one would be the rectangle in the center, and one would be the triangle on the right-hand side. This proof works just fine for the standard-model trapezoid, where the shorter base is on top, and it’s centered above the longer base. It also works fine if the center of the shorter base is directly above the right end of the lower base, but it falls apart in other cases: if you have a right trapezoid, for example, there’s only the one triangle to lop off. If the upper base isn’t anywhere above the lower base, the proof doesn’t work, but we could repair it.

That inspires the differences that I see, though: How many of the ends of the upper base are above the lower base? And put that way, there are only three possible answers. Either both ends of the upper base are above the lower base, or just one end of the upper base is above the lower, or else neither end of the upper base is above the lower. If both ends of the upper base aren’t above the lower base, they might be either to the left or to the right of the lower base; I’ll draw them going over to the right side. This implies there are three kinds of trapezoids; let me show how we get more of them.

This is the first case with both ends of the upper base above the lower base. And this suddenly shows how to make six cases out of these three possibilities: the ends of the upper base are directly above the ends of the lower base. This is the shape we might more commonly call a rectangle, and we can argue about whether rectangles are trapezoids, but let’s suppose they are or the number of possibilities drops to five.

This is the second case of having both ends of the upper base above the lower base. One end is directly above an end of the lower base; the other is between the ends of the lower base. This is the right trapezoid shape so handy in working out the rules for integration. This is the only example of a right trapezoid that we need, since it wouldn’t really be different if the vertical leg were on the right-hand side rather than the left. And we’re taking by assumption the longer base to be on the bottom, so we don’t need to consider the case of the longer base being above the shorter.

This is the third example of both ends of the shorter base being above the longer base, and it’s also the standard-issue trapezoid, the one shown in textbooks when writing out formulas for areas. This is also one where the area formula could be found by chopping the segment up into three pieces. Do it by dropping vertical lines at the points where the upper base has its ends. You get two triangles and a rectangle.

Figuring out the area of the rectangle should be easy — obvious even — although figuring out the area of each of the triangles may be a bit mysterious. After all, you need the base and the height of each triangle to find its area. The height is obvious; it’s how far apart the parallel bases are. The width, though … that seems to depend heavily on the angle the legs make. It’s worth thinking about how to find the area of these two triangles.

This is the first example of a trapezoid where the upper base has only one end point above the lower base. The right end of the top stretches out past the right end of the bottom. (It wouldn’t be different if the upper base stretched out past the left end instead.) This is probably the second-most-obvious case of a trapezoid at all. The area formula could also be worked out by chopping it up along vertical lines, slicing the trapezoid at the left end of the upper base and at the right end of the lower base.

There’s the same problem in figuring out the areas of both of these triangles, since there’s so much variability in the bases of the two triangles. But if you spotted how to work out the area of the two triangles above then this won’t give you any trouble.

Here’s the second example of just the one end of the upper base being above the lower base. It may not look very different from the previous one, but the difference shows up if you try slicing vertically at the upper-left or the lower-right ends. The left end of the upper base is directly over the right end of the lower base, and so, there’s just the one slice, and it cuts the trapezoid into two triangles, with no rectangle.

This is the example where neither end of the upper base is above the lower base. I don’t see any obvious special cases to this, unless you want to count where the two legs happen to be parallel and make the trapezoid into a parallelogram. If you do try cutting this with vertical slices at the left end of the upper base and the right end of the lower base you come out of it with two short triangles and a brand-new trapezoid, which is just a mess to deal with.

There might be another case worth considering: what about the parallelogram, where the upper base is far enough to the right of the lower base that no point of it is above the lower? And I think we don’t need to pay any special attention to this, because if we rotate that figure a quarter-turn, making what had been the two legs into the bases, then we have a parallelogram with one end of the upper base above the lower base, which I claim we already covered with the fourth kind of trapezoid.

Of course, I claim this, but does that mean I’m right? It may be fun to spend some time convincing yourself of this, or finding a counter-example. Perhaps a seventh kind needs to be added after all.

## Everything I Know About Trapezoids | nebusresearch 12:29 am

onTuesday, 17 April, 2012 Permalink |[…] Responses How Many Trapezoids … on How Many Trapezoids Can You…Joseph Nebus on How Do You Make A Trapezoid…Joseph […]

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## Something I Didn’t Know About Trapezoids | nebusresearch 12:15 am

onFriday, 20 April, 2012 Permalink |[…] Everything I Know Ab… on How To Forget The Area Of A…Everything I Know Ab… on How Many Trapezoids I Can…Everything I Know Ab… on How Many Trapezoids Can You…Everything I Know Ab… on […]

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## My June 2013 Statistics | nebusresearch 1:48 pm

onMonday, 1 July, 2013 Permalink |[…] to find their area, and how many kinds of trapezoids there are — with such catchy titles as How Many Trapezoids I Can Draw, or How Do You Make A Trapezoid Right?, or Setting Out To Trap A Zoid, which should be recognized as […]

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## My July 2013 Statistics | nebusresearch 7:03 pm

onThursday, 1 August, 2013 Permalink |[…] all-time most popular post remains How Many Trapezoids I Can Draw, because I think there are people out there who worry about how many different kinds of trapezoids […]

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## September 2013′s Statistics | nebusresearch 12:25 am

onTuesday, 1 October, 2013 Permalink |[…] How Many Trapezoids I Can Draw […]

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## October 2013′s Statistics | nebusresearch 10:09 pm

onFriday, 1 November, 2013 Permalink |[…] How Many Trapezoids I Can Draw […]

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## November 2013′s Statistics | nebusresearch 8:48 pm

onSunday, 1 December, 2013 Permalink |[…] How Many Trapezoids I Can Draw […]

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## nebusresearch | December 2013′s Statistics 5:22 pm

onSaturday, 4 January, 2014 Permalink |[…] How Many Trapezoids I Can Draw, showing that my whole trapezoid-area fiasco was maybe the best thing that ever happened to me. […]

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## ivasallay 8:20 pm

onFriday, 24 January, 2014 Permalink |Reblogged this on Find the Factors and commented:

A trapezoid is often defined for young students as a four-sided shape with EXACTLY two parallel sides. Once a person studies higher level math, the definition changes: A trapezoid is a four-sided shape with AT LEAST two parallel sides. How many different kinds of trapezoid can a person draw? It depends on which definition you use. If you use the second definition, you can also include parallelograms, rectangles, rhombuses, and squares. Either definition will allow the standard isosceles trapezoid and several others. But how many? Whichever definition you use, figuring out how many different ones can be drawn is a nice puzzle to solve. This blog post does a nice job explaining the different ones, and it even came up with ones I hadn’t considered!

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## nebusresearch | January 2014′s Statistics 4:57 pm

onSaturday, 1 February, 2014 Permalink |[…] How Many Trapezoids I Can Draw, with my best guess for how many different kinds of trapezoids there are (and despite its popularity I haven’t seen a kind not listed here, which surprises me). […]

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## nebusresearch | February 2014′s Mathematics Blog Statistics 12:28 am

onSaturday, 1 March, 2014 Permalink |[…] How Many Trapezoids I Can Draw, which will be my immortal legacy. […]

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## nebusresearch | 14,000 7:00 pm

onSaturday, 8 March, 2014 Permalink |[…] to show why the area of a trapezoid was the trapezoid formula. My most-read article of all time is How Many Trapezoids I Can Draw, which is still standing at six by the way; some of the other articles which went into that (like […]

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## nebusresearch | The Math Blog Statistics, March 2014 12:09 am

onTuesday, 1 April, 2014 Permalink |[…] How Many Trapezoids I Can Draw, and again, nobody’s found one I overlooked. […]

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## nebusresearch | The Math Blog Statistics, April 2014 1:35 pm

onThursday, 1 May, 2014 Permalink |[…] How Many Trapezoids I Can Draw, always with the trapezoids […]

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## nebusresearch | The Math Blog Statistics, May 2014 3:42 pm

onMonday, 2 June, 2014 Permalink |[…] How Many Trapezoids I Can Draw, which will be my memorial […]

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## nebusresearch | My Math Blog Statistics, September 2014 4:01 pm

onWednesday, 1 October, 2014 Permalink |[…] How Many Trapezoids I Can Draw, which, at the risk of spoiling its popularity, I figure to be: six. […]

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## nebusresearch | My Math Blog Statistics, October 2014 2:45 pm

onSaturday, 1 November, 2014 Permalink |[…] How Many Trapezoids I Can Draw, my best bid for mathematical immortality. […]

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## nebusresearch | My Math Blog Statistics, November 2014 1:34 am

onMonday, 1 December, 2014 Permalink |[…] How Many Trapezoids I Can Draw (six that are any different, […]

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## nebusresearch | 20,000: My Math Blog’s Statistics 5:46 pm

onFriday, 2 January, 2015 Permalink |[…] How Many Trapezoids I Can Draw, my life’s crowning achievement. (Six. If you find a seventh please let me know and I’ll do a follow-up post.) […]

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## How My Mathematics Blog Was Read, For January 2015 | nebusresearch 8:15 pm

onSunday, 1 February, 2015 Permalink |[…] How Many Trapezoids I Can Draw, which also reveals how many trapeziums I think are different in interesting ways. […]

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## How February 2015 Treated My Mathematics Blog | nebusresearch 9:52 pm

onSunday, 1 March, 2015 Permalink |[…] How Many Trapezoids I Can Draw, which is the closest I’ll come to classifying the sporadic finite simple groups. […]

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## My Mathematics Blog, As March 2015 Would Have It | nebusresearch 7:33 pm

onWednesday, 1 April, 2015 Permalink |[…] How Many Trapezoids I Can Draw, of course. […]

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## sheldonk2014 10:44 pm

onSunday, 12 April, 2015 Permalink |I like this post because I was just trying a different approach to my writing today now of course you speaking about something totally different but in my mind the process is some what the same,how do you think outside the box to arrive at the answer,so it took some doing but I took a bare bone idea that I had written a turned it on its ear,you took a shape and wanted to see how many more,to me it’s the same,

As always Sheldon

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## Joseph Nebus 4:06 am

onFriday, 17 April, 2015 Permalink |Well, ah, thank you, I believe. I don’t know that I was trying anything particularly outside-the-box when I tried thinking how many different trapezoids there were. It amounted to more thinking about what sorts of things stand out to me as trapezoids, and how I might describe them to someone who wasn’t looking at the same picture I was.

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## sheldonk2014 11:52 pm

onSunday, 12 April, 2015 Permalink |This is the one I was speaking about,how I took it and reworked it

Thank you for visiting

As always Sheldon

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## Joseph Nebus 4:17 am

onFriday, 17 April, 2015 Permalink |Ah, thank you.

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## How April 2015 Treated My Mathematics Blog | nebusresearch 3:51 pm

onFriday, 1 May, 2015 Permalink |[…] How Many Trapezoids I Can Draw. […]

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## Reading the Comics, May 9, 2015: Trapezoid Edition | nebusresearch 8:13 pm

onSaturday, 9 May, 2015 Permalink |[…] Poland’s Robbie and Bobby (May 6) is trying to worm its way into my “How Many Trapezoids I Can Draw” post. Little does the Shape Shifter know that I already counted squares as a kind of trapezoid. Well, I […]

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## How May 2015 Treated My Mathematics Blog | nebusresearch 8:00 am

onTuesday, 2 June, 2015 Permalink |[…] Playful Mathematics: Sweet Add-A-Line […]

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## How October Treated My Mathematics Blog | nebusresearch 4:00 pm

onMonday, 2 November, 2015 Permalink |[…] How Many Trapezoids I Can Draw […]

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## How November 2015 Treated My Mathematics Blog | nebusresearch 3:01 pm

onWednesday, 2 December, 2015 Permalink |[…] How Many Trapezoids I Can Draw […]

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## dan 11:16 am

onSaturday, 21 May, 2016 Permalink |this was interesting but i find myself wondering, what is the underlying rule that is being used to define what makes a different type. i feel like there is one, but not sure how to express it mathematically.

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## Joseph Nebus 3:30 am

onThursday, 2 June, 2016 Permalink |This is a good question and I’m not sure how to characterize it exactly. I may have to write a follow-up post to say how I came to figure on these trapezoids rather than another set.

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## How April 2017 Treated My Mathematics Blog | nebusresearch 6:03 pm

onTuesday, 9 May, 2017 Permalink |[…] How Many Trapezoids I Can Draw and nobody’s found one I’ve missed yet! […]

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