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Area of Quadrilaterals

Now we can talk about areas of quadrilaterals. This video will review, how to find the area of the special quadrilaterals. So first of all, the most special quadrilateral is a square. If a square has a side of s, then the area is simply s squared. In fact, even in algebra, raising a number to the second power is, is called squaring precisely because, this is the way to find the area of a square.

So the word we use in algebra, actually comes, from the basic geometric fact. For the other special quadrilaterals, the general formula, is Area = bh, but we need to think carefully about this. The formula A = bh, most obviously works for a rectangle, in which b and h are sides of the rectangle. As with triangles, remember that the, base, needs not be horizontal.

Any side can be the base, and the height must be perpendicular to it. So for a rectangle the area’s just the product of the two different side lengths. The area A = bh, also works for rhombuses and parallelograms and any side can be a base, but the height has to be perpendicular, to the base so the height will not lie along a side. Instead, the height will be what’s called an altitude aligned perpendicular to the base, and so we need to find that height.

The length of the altitude is not given, then almost always one can find it from the Pythagorean theorem. So here’s a practice problem. Pause the video and then we’ll talk about this. Okay. So if JN = 1 and NM = 2, then all the way across from J to M has to be 3.

And because it’s a rhombus, every side has to have a length of 3. So, now look at the right triangle JKN. In which, JK, the side of the rhombus is 3, and JN were given as 1. We’ll use the Pythagorean Theorem in that triangle to find KN. KN squared equals JK minus JN squared. 3 squared is 9, so 9 minus 1 is 8.

Means that KN is the square root of 8. Of course we can simplify that down to 2 root 2. And that is the height of the rhombus. So now we are ready to apply, area equals one half, equals base times height. We know that the base, is 3 and the height is 2 root 2. And we can simply multiple these and get 6 root 2.

If these operations with roots are a little bit unfamiliar, I would suggest going back to the power and roots module and watching the video, operations with roots. With trapezoids, we have to re-think a bit, because there are two bases, two parallel sides. So, what exactly would we mean by base times height.

Well the height is pretty clear, but we have two bases, so what are we gonna do? One way to find the area, is to find the average of the bases, and multiply this by the height. So that is the formula, for the area of a rhombus. We average the bases and multiply the height times the average of the bases. Sometimes we can find the area of a trapezoid by subdividing the trapezoid, into a central rectangle and two side right triangles.

So this is often what the test will have us do. We have two side right triangles, we can find information about those, with the Pythagorean theorem. And that will allow us to solve for everything and find all the areas. And of course, if it’s a symmetrical trapezoid, those two sides would become congruent, which makes things even easier.

Here’s a practice problem, pause the video and then we’ll talk about this. In trapezoid ABCD, altitudes are drawn. If AF = 5, find the area of the trapezoid. Well, first of all, we’re gonna look at ABF. That little triangle, we have a leg of 5 an unknown leg and a hypotenuse of 13.

So of course that’s a 5, 12, 13 triangle. And BF equals 12. So we can find that just from knowledge of our Pythagorean triplet. So we don’t even need to do a calculation. So BF equals 12. That means that CE also equals 12.

So, you can find the area of the triangle ABF and that has to be one half five times 12. One half six. One half, sorry. Five times six is 30. You can also find the area of the rectangle 10 times 12 is a 120.

Now, that triangle on the left, triangle CED, that’s going to be blank, 12, 15. Well, of course, that’s a {3, 4, 5} triangle, multiplied by 3. So, that’s going to be {9, 12, 15}. And then, the area, one-half 9 times 12. Well, that’s 9 times 6 which is 54. So the whole area is gonna be triangle plus rectangle plus triangle, 30 plus 120 plus 54, and that equals 204.

In summary, a square has an area of s squared. The rectangle, rhombus, and parallelogram have an area of base times height. We have to be careful in a rhombus or a parallelogram. Any side can be the base but the other side is not gonna be the height. The height has to be perpendicular to the base. We need to find the altitude.

A trapezoid is the average of the bases times the height. And for any slanty shapes. Think about subdividing into rectangles and right triangles, and this might even be true, for example, if we were dealing with an irregular quadrilateral. And expect to find the Pythagorean theorem involved in anything involving a slant