Back To CourseGeometry: High School
15 chapters | 160 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
First, let's go over what a rhombus is. A rhombus is a four-sided shape whose sides are all equal and whose opposite sides are parallel. If you are given the measurement of just one side, then you also know the measurement of all the other sides since they are all equal.
A rhombus has five additional measurements that we can consider. I've marked them down on the rhombus below so you can see them. If you look at the red dashed line going straight up and down, that is the altitude, or the height of the rhombus. It is not how long the side is but how high the rhombus is if it was sitting on a flat surface. Note how the bottom side is flat. The next two measurements are the diagonals, the lines connecting the opposite angles to each other. I've drawn the diagonals using blue dashed lines. Do you see them? I've labeled one diagonal p and the other q so you know which is which.
The red s is the measurement for the length of a side. If one side is marked s, then all the other sides are also s because all the sides of a rhombus are equal in length to each other. The fourth and fifth measurements we can note are the measurements of the angles. We can call one pair of opposite angles Angle A and the other opposite pair of angles Angle B. It doesn't matter how you label these as long as you label one pair one name and the other pair another name.
Now that we have all of our various measurements that we can consider, let's go over the three different ways to find the area of a rhombus. I encourage you to commit these three formulas to memory. Use flashcards or whatever memory aid that helps you.
One way to find the area of a rhombus is by using the altitude and the side. If we know these two measurements, then we can use this method. The formula for this method is:
Area = Altitude * s
Looks pretty simple, right? All you have to do is to multiply the altitude with one of the sides. It doesn't matter which side, as they're all the same.
So, if you had a rhombus whose altitude is 6 inches with sides that are 2 inches, then the area of this rhombus is 6 inches * 2 inches = 12 inches squared.
Now, if instead of the altitude, you are given one of the angle measurements, you can use another method for finding the area of a rhombus. This method uses the side measurement and one of the angle measurements, and the formula looks like this:
Area = s^2 sin (A) or Area = s^2 sin (B)
The formula is the same regardless of which angle you are given. All you need is the measurement of one of the angles and the side. You also multiply here, but first you have to find the sine of the angle as well as squaring your side. This formula is a bit harder to memorize than the one using the altitude and side, but it is still a good idea to commit this to memory. Write it down on a flashcard if you need to.
If our rhombus tells us that our side measures 2 inches and one of our angles measures 60 degrees, then to find the area of this rhombus, we would plug this into our formula for area using the side and an angle.
Area = 2^2 sin (60)
We square our side, and find the sine of 60 degrees. So:
Area = 4 * 0.866
Then, we multiply these two numbers together to get our answer.
Area = 3.46 inches squared
I want you to remember that for this method the angle you choose does not matter. The formula is the same for both angles; you just have to pick one of them to use.
The last way of finding the area of a rhombus uses only the two diagonals. The formula looks like this:
Area = (p * q)/2
We are multiplying the two diagonals together and finding half of that. Again, for this formula, it does not matter how you label the diagonals as long as one is one measurement and the other is the other measurement. Of course, if p and q are already specified, then stick with that. But, if they haven't been, then you can label either one p and the other will be q.
If we are given a rhombus with only the measurements for the diagonals, we would use this formula. If our rhombus had diagonals measuring 3 inches and 4 inches respectively, then to find the area of this rhombus, we would do the same as we've done with the other formulas and plug the numbers in.
Area = (3 * 4)/2
We multiply the two diagonals, and:
Area = 12/2
We then find half of that, so:
Area = 6 inches squared
And this is our answer.
We've learned in this video lesson that finding the area of a rhombus, a four-sided shape whose sides are all equal and whose opposite sides are parallel, is quite easy when you follow the formulas. The five measurements to consider are altitude or the height of the rhombus, the two diagonals, the length of a side, and the measurement of the two pairs of opposite angles.
Do your best to memorize the following three formulas:
The formula for the altitude and side is Area = Altitude * s.
The formula for the side and an angle is Area = s^2 sin (A) or Area = s^2 sin (B).
The formula with the diagonals is Area = (p * q)/2.
Use whichever equation you have the information for. Using the formulas is a matter of plug and play when you know all the numbers to plug in, so it's pretty straightforward. Just commit to memory what the formulas look like.
After this lesson, you'll be able to:
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Back To CourseGeometry: High School
15 chapters | 160 lessons