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### Course: High school geometry>Unit 3

Lesson 6: Theorems concerning quadrilateral properties

# Proof: Rhombus area

Sal proves that we can find the area of a rhombus by taking half the product of the lengths of the diagonals. Created by Sal Khan.

## Want to join the conversation?

• in a rhombus, are the angles alwyas right angles?
(15 votes)
• No, a rhombus only means that all the sides are of equal length. A Square, is a Rhombus where all the sides intersect at right angles.
The diagonals of a Rhombus, however, do always intersect at right angles.
(66 votes)
• How do you remember the meaning of all the shapes? For example:
- Rhombus
- Kite
- Square
- Rectangle
and so on...
(5 votes)
• Try to link the shapes with what you know and your environment. Like, a rhombus looks like someone had sat on a square slightly, the kite would be like a square but tilted forward while a square is taking two 'L's together and putting them on top on each other.
(12 votes)
• The plural of rhombus is rhombi, but how does one know which words to say "uses" and which words to say "i"? I've always wondered that.
(7 votes)
• Good question. It will depend of which language the word is taken from. Rhombus comes from the greek rhombos. It was then used in latin as rhombus. The plural of a -us word in latin is -i. Sometimes, a foreign word will be anglicized and it will lose its weird (for us) plural form. Language, like mathematics, is conventions that were agreed upon at one point in history and is not always logical. ;)
(7 votes)
• But what if it asked a question like "Find the area of rhombus when a side and diagonal are given"
Like when a diagonal is 12 and side is 10 ?
Please help me to find this..
(4 votes)
• Since the diagonals of a rhombus are perpendicular, you can use the Pythagorean theorem to find the other diagonal and then find the area.
(8 votes)
• whats the difference between perpendicular and parelel?
(5 votes)
• For example parallel lines have the same slope and different y intercepts, while a perpendicular line is the negative reciprocal of the other slope.In shapes now parallel lines are basically opposite of each other and don't intersect.While perpendicular lines always for right angles or 90 degree angles.
(3 votes)
• If it is already so completely evident that the rhombus is congruent, why must we go through all these steps of trying to prove it? Also, I still don't understand how to use the different congruency methods, like SSS, SAS, ASA, and AAS. Can someone please explain how and when to use each of them?
(3 votes)
• This proof that Sal demonstrates is called two-column proof. He is not writing all the steps since he has already given us the steps by word. However, the two-column proof is the basis of proof in geometry, and it is what you use to explain your actions in a problem (as Sal did two videos ago).

The Postulates
Before reading the definitions of these postulates, you should realize that these specific ones are strictly for triangles!

SSS- Side-Side-Side postulate is one method to prove a triangle is congruent to another. You can use this when two triangles have been discovered to share the same three side lengths.

SAS- Side-Angle-Side postulate is another method to prove that one triangle is congruent to another. You can use this when two triangles have been found to share two side lengths. Along with that, one angle that lies between those side lengths is found to have the same measurement as its corresponding counterpart on the other triangle.

ASA- Angle-Side-Angle postulate is the converse of Side-Angle-Side postulate, and is another method you can use to prove two triangles are congruent. ASA is when two angles are found to have shared corresponding counterparts measures on both triangles. Along with that, one side in between those angles is found to have the same length as its corresponding side on the other triangle.

AAS- Angle-Angle-Side postulate is the final method to prove that a triangle is congruent to another. AAS, is very similar to ASA, however it has one key difference. AAS is when you find two angles to be the same measure as their corresponding counterpart on the other triangle. However, in this postulate, the side that is found to be the same length as its corresponding counterpart is not in between the angles; it's next to one of the angles but not the other.

Hope this helps
(6 votes)
• Do those brackets Sal uses indicate that you are finding the area? In other words, does writing "[ADC]" mean "the area of ADC".
(2 votes)
• Where did you get the 1\4 in the rhombus area video
(4 votes)
• Sal says that [ABC] is equivalent to 0.5*AC*0.5*BD and that is equal to 0.25*AC*BD, since 0.5*0.5 is equal to 0.25
(3 votes)
• I have a math question on an assignment that is asking this sort of thing, but it calls the shape a parallelogram instead of a rhombus, does the name matter? Could the steps in figuring it out stay the same? Thanks sal...
(3 votes)
• A rhombus is a special type of parallelogram, one where all four sides are congruent. The proof given here depends on the fact that all four sides are congruent, so if that isn't the case for your figure, the proof will fail.

It's also the case that the statement in this video is false for general parallelograms. Consider a rectangle with a length of 1 and width of 2. The area of the rectangle is 2, but half the product of the diagonals is 5/2, not 2.
(3 votes)
• This is kind of confusing, can somebody explain this in one sentence. Thank you!
(3 votes)
• To find the area of a rhombus you have to use the same formula as a parallelogram which is A= BxH
(3 votes)

## Video transcript

So quadrilateral ABCD, they're telling us it is a rhombus, and what we need to do, we need to prove that the area of this rhombus is equal to 1/2 times AC times BD. So we're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals. So let's see what we can do over here. So there's a bunch of things we know about rhombi and all rhombi are parallelograms, so there's tons of things that we know about parallelograms. First of all, if it's a rhombus, we know that all of the sides are congruent. So that side length is equal to that side length is equal to that side length is equal to that side length. Because it's a parallelogram, we know the diagonals bisect each other. So we know that this length-- let me call this point over here B, let's call this E. We know that BE is going to be equal to ED. So that's BE, we know that's going to be equal to ED. And we know that AE is equal to EC. We also know, because this is a rhombus, and we proved this in the last video, that the diagonals, not only do they bisect each other, but they are also perpendicular. So we know that this is a right angle, this is a right angle, that is a right angle, and then this is a right angle. So the easiest way to think about it is if we can show that this triangle ADC is congruent to triangle ABC, and if we can figure out the area of one of them, we can just double it. So the first part is pretty straightforward. So we can see that triangle ADC is going to be congruent to triangle ABC, and we know that by side-side-side congruency. This side is congruent to that side. This side is congruent to that side, and they both share a C right over here. So this is by side-side-side. And so we can say that the area-- so because of that, we know that the area of ABCD is just going to be equal to 2 times the area of, we could pick either one of these. We could say 2 times the area of ABC. Because area of ABCD-- actually let me write it this way. The area of ABCD is equal to the area of ADC plus the area of ABC. But since they're congruent, these two are going to be the same thing, so it's just going to be 2 times the area of ABC. Now what is the area of ABC? Well area of a triangle is just 1/2 base times height. So area of ABC is just equal to 1/2 times the base of that triangle times its height, which is equal to 1/2. What is the length of the base? Well the length of the base is AC. So it's 1/2-- I'll color code it. The base is AC. And then what is the height? What is the height here? Well we know that this diagonal right over here, that it's a perpendicular bisector. So the height is just the distance from BE. So it's AC times BE, that is the height. This is an altitude. It intersects this base at a 90-degree angle. Or we could say BE is the same thing as 1/2 times BD. So this is-- let me write it. This is equal to, so it's equal to 1/2 times AC, that's our base. And then our height is BE, which we're saying is the same thing as 1/2 times BD. So that's the area of just ABC, that's just the area of this broader triangle right up there, or that larger triangle right up there, that half of the rhombus. But we decided that the area of the whole thing is two times that. So if we go back, if we use both this information and this information right over here, we have the area of ABCD is going to be equal to 2 times the area of ABC, where the area of ABC is this thing right over here. So 2 times the area of ABC, area of ABC is that right over there. So 1/2 times 1/2 is 1/4 times AC times BD. And then you see where this is going. 2 times 1/4 fourth is 1/2 times AC times BD. Fairly straightforward, which is a neat result. And actually, I haven't done this in a video. I'll do it in the next video. There are other ways of finding the areas of parallelograms, generally. It's essentially base times height, but for a rhombus we could do that because it is a parallelogram, but we also have this other neat little result that we proved in this video. That if we know the lengths of the diagonals, the area of the rhombus is 1/2 times the products of the lengths of the diagonals, which is kind of a neat result.