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Area of Quadrilaterals


Now we can talk about areas of quadrilaterals. This video we will review how to find the area of the special quadrilaterals. So first of all the most special quadrilaterals is a square, if a squares the side of s, then the area is simply s squared. In fact, even an algebra, raising a number 2, the second power 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. The other special quadrilaterals, the general formula is area equals base times height, but we need to think carefully about this. The formula area equals base times height most obviously works for a rectangle, in which b and h or 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 is 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 a line 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 than 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 we are given as 1. We'll use the Pythagorean theorem in that triangle to find KN. KN squared equals JK- JN squared, 3 squared is 9, so 9- 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're ready to apply area equals equals base times height. We know that the base is 3 and the height is 2 root 2, and we can simply multiply 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, we average the bases and multiply the height times the average of the base. Sometimes we can find the area of a trapezoid by subdividing the trapezoid into a central rectangle into side right triangles. So this is often what the test will have us do. We have to 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 episode, those two side right triangles will be 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 a 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 = 12. So we can find that just from knowledge of our Pythagorean triplet, we don't even need to do a calculation. So BF = 12, that means that CE also equals 12.

So we can find the area of the triangle ABF and that has to be one half, five times 12 one half six, 5ive times 6 is 30. We can also find the area of the rectangle 10 time 12 is 120. That triangle on the left, triangle CED, that's gonna be blank 12, 15. Well, of course, that's a 3, 4, 5 triangle multiplied by 3, so that's gonna be 9, 12, 15.

And then the area, 1 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 + 120 + 54 and that equals 204. In summary, a square has an area of s squared. The rectangle, rhombus and parallelogram has an are of base times height, we have to 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're dealing with any irregular quadrilaterals.

And expect to find the Pythagorean theorem involved in anything involving a slant.

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