Area of a Trapezoid
The formula for the area of a trapezoid is breathtakingly easy to understand when we slice and dice a picture to show why the formula works.
But most basal texts, when presenting this topic—and a number of other geometry topics involving formulas—use just one picture or maybe just a few different pictures to get the idea across. And doing so leaves open the question of whether the presentation indeed covers all examples of trapezoids or cylinders or parallelograms or what have you.
This is, in some sense, understandable. There is a limited amount of real estate in any basal text, and a more general (more accurate) explanation may make the topic much more difficult to understand. If basals try at all to explore an "all cases" type of explanation, they tend to do so with hands-on activities—a strategy that can be useful (especially with the presentation below) but is more often just an expansion of the number of cases that can be explored.
Let's take a crack at explaining—to adults like you—the formula for the area of all trapezoids: 1/2 h(b1 + b2), and maybe someone out there can improve upon it and/or develop a nifty hands-on complement to it.
We'll use the most typical definition for trapezoid: A quadrilateral that has exactly one pair of parallel sides. You can see the parallel sides and the non-parallel sides on the trapezoid below.
There are three important parts of a trapezoid: the bottom base, labeled b1, the top base, labeled b2, and the height, labeled h. You can see all of these in the figure.
The two dotted line segments—which include the non-parallel sides of the trapezoid—are fixed to the bottom horizontal line. You can imagine that both of these line segments can slide left or right and/or rotate on the points shown, stretching or shrinking the top and/or bottom of the trapezoid as they move and changing the angle(s) of one or both of the non-parallel sides. For a little warm-up you can mentally play with this idea in the figure above, sliding and swiveling the non-parallel sides. We don't mess with the parallel sides, though, because "exactly one pair of parallel sides" is essential for "trapezoid-ness." If you like, you can imagine rotating the parallel lines together, so they are not horizontal. As long as they are parallel, this won't affect our explanation.
Let's draw a line inside the trapezoid above that will separate it into two triangles.
You can see that both of the triangles have the same perpendicular height, h. What's interesting is that one of the triangles has b1 as its base, and the other has b2 as its base. And this is key: We want to show that ANY trapezoid can be separated into two triangles—one with a base of b1, and the other with a base of b2--that both have the same height.
Okay, so let's swivel the dotted line on the right outward and leave the dotted line on the left alone.
By doing this, we stretch just the top of the trapezoid (b2). We also stretch that red line that divides the figure into two triangles. Yet, as you can see, we still have two triangles with the same height, and one triangle has a base of b1, while the other has a base of b2.
So far, so good. Now let's swivel and slide both of the dotted lines outward.
Here we've stretched the top and bottom of the trapezoid along with the red dividing line, but we still have two triangles with the same height, one with a base of b1 and the other with a base of b2.
It is possible to swivel and slide our way (inward) into a triangle, a figure that obviously does not meet our definition of trapezoid. Similarly, when we swivel both dotted lines out far enough, we create a parallelogram (in this case, a rectangle), which also does not meet our definition of trapezoid, because then the figure would have two pairs of parallel sides. (The formula still works for parallelograms. Since the two bases are congruent, the formula becomes 1/2h(2b), which simplifies to bh.)
And, in fact, we can stop there—we have shown (with the help of your imagination) that every possible trapezoid can be divided into two triangles with the same height, one with a base of b1 and the other with a base of b2.
But, you might ask, what about continuing the outward swiveling of our dotted lines? We can continue to rotate them outward, creating different trapezoids.
The reason we don't need to deal with these examples is because they are all inversions—basically, upside-down cases—of all the possibilities we have already seen. The image below is, for example, an inversion of the second image I showed above.
And, of course, as I mentioned, it doesn't matter how we rotate our parallel lines. The same explanation applies.
So, every trapezoid can be separated into two triangles with the same height—one with a base of b1 and the other with a base of b2.
To find the area of a trapezoid, then, we add the areas of the triangles. The area of one of the triangles is 1/2b1h, and the area of the other is 1/2b2h. If we add these together, we get 1/2b1h + 1/2b2h.
And a little fancy Distributive Property pencilwork reveals 1/2h(b1 + b2).
But most basal texts, when presenting this topic—and a number of other geometry topics involving formulas—use just one picture or maybe just a few different pictures to get the idea across. And doing so leaves open the question of whether the presentation indeed covers all examples of trapezoids or cylinders or parallelograms or what have you.
This is, in some sense, understandable. There is a limited amount of real estate in any basal text, and a more general (more accurate) explanation may make the topic much more difficult to understand. If basals try at all to explore an "all cases" type of explanation, they tend to do so with hands-on activities—a strategy that can be useful (especially with the presentation below) but is more often just an expansion of the number of cases that can be explored.
Let's take a crack at explaining—to adults like you—the formula for the area of all trapezoids: 1/2 h(b1 + b2), and maybe someone out there can improve upon it and/or develop a nifty hands-on complement to it.
We'll use the most typical definition for trapezoid: A quadrilateral that has exactly one pair of parallel sides. You can see the parallel sides and the non-parallel sides on the trapezoid below.
There are three important parts of a trapezoid: the bottom base, labeled b1, the top base, labeled b2, and the height, labeled h. You can see all of these in the figure.
The two dotted line segments—which include the non-parallel sides of the trapezoid—are fixed to the bottom horizontal line. You can imagine that both of these line segments can slide left or right and/or rotate on the points shown, stretching or shrinking the top and/or bottom of the trapezoid as they move and changing the angle(s) of one or both of the non-parallel sides. For a little warm-up you can mentally play with this idea in the figure above, sliding and swiveling the non-parallel sides. We don't mess with the parallel sides, though, because "exactly one pair of parallel sides" is essential for "trapezoid-ness." If you like, you can imagine rotating the parallel lines together, so they are not horizontal. As long as they are parallel, this won't affect our explanation.
Let's draw a line inside the trapezoid above that will separate it into two triangles.
You can see that both of the triangles have the same perpendicular height, h. What's interesting is that one of the triangles has b1 as its base, and the other has b2 as its base. And this is key: We want to show that ANY trapezoid can be separated into two triangles—one with a base of b1, and the other with a base of b2--that both have the same height.
Okay, so let's swivel the dotted line on the right outward and leave the dotted line on the left alone.
By doing this, we stretch just the top of the trapezoid (b2). We also stretch that red line that divides the figure into two triangles. Yet, as you can see, we still have two triangles with the same height, and one triangle has a base of b1, while the other has a base of b2.
So far, so good. Now let's swivel and slide both of the dotted lines outward.
Here we've stretched the top and bottom of the trapezoid along with the red dividing line, but we still have two triangles with the same height, one with a base of b1 and the other with a base of b2.
It is possible to swivel and slide our way (inward) into a triangle, a figure that obviously does not meet our definition of trapezoid. Similarly, when we swivel both dotted lines out far enough, we create a parallelogram (in this case, a rectangle), which also does not meet our definition of trapezoid, because then the figure would have two pairs of parallel sides. (The formula still works for parallelograms. Since the two bases are congruent, the formula becomes 1/2h(2b), which simplifies to bh.)
And, in fact, we can stop there—we have shown (with the help of your imagination) that every possible trapezoid can be divided into two triangles with the same height, one with a base of b1 and the other with a base of b2.
But, you might ask, what about continuing the outward swiveling of our dotted lines? We can continue to rotate them outward, creating different trapezoids.
The reason we don't need to deal with these examples is because they are all inversions—basically, upside-down cases—of all the possibilities we have already seen. The image below is, for example, an inversion of the second image I showed above.
And, of course, as I mentioned, it doesn't matter how we rotate our parallel lines. The same explanation applies.
So, every trapezoid can be separated into two triangles with the same height—one with a base of b1 and the other with a base of b2.
To find the area of a trapezoid, then, we add the areas of the triangles. The area of one of the triangles is 1/2b1h, and the area of the other is 1/2b2h. If we add these together, we get 1/2b1h + 1/2b2h.
And a little fancy Distributive Property pencilwork reveals 1/2h(b1 + b2).
Labels: mathematics, textbooks