Geometry: Area of Quadrilaterals

Do you know how to find the area of quadrilaterals? Well, in this video I will show you how.

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Transcript: Area of Quadrilaterals

Now we can talk about the area of quadrilaterals. In this video, we will review how to find the area of the special quadrilaterals.

The Square

So first of all the most special quadrilateral is the square. If a square has the side of s, then the area is simply s squared. In fact, even in algebra, raising a number to 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 a basic geometric fact. For 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–works most obviously for a rectangle, in which b & h are both sides of the rectangle.

Triangles

And 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.

Practice Problem

So here is 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, a nd JN were given as 1. We’ll you use the Pythagorean theorem in that triangle to find KN. KN squared = (JK)- (JN) squared. 3 squared is 9, so 9- 1=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 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 this operation 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.

Trapezoids

With trapezoids, we have to re-think a bit, because there are two bases, two parallel sides.

area of quadrilaterals, acute trapezoid - magoosh

ACUTE TRAPEZOID
Image by icon99

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 base.

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 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 ABF.

That little triangle we have a leg of 5, an unknown leg, and a hypotenuse of 13. 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.

We can find the area of the triangle ABF, and that has to be one half 5 times 12. 5 times 6 is 30. We can also find the area of the rectangle, 10 times 12 is a 120. Now 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 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 + 120 + 54, and that equals 204.

Summary of How to Find the Area of Quadrilaterals

In summary, a square has an area of s squared. The rectangle, rhombus, parallelogram have an area of base times height.

We have to be careful in a rhombus or 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 slanted 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.

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