Question

15. Two sides of a rhombus form a 120º angle. The length of each side is 6 in. Explain how to find the area of the rhombus, and then calculate the area. Leave your answer in simplest radical form.

Answer 1

Definition of a rhombus, all 4 sides the same length. Right?

Answer 2

Looks like a diamond
http://en.wikipedia.org/wiki/Rhombus

Answer 3

What else do we know?

Answer 4

two side of the angle form 120 degrees

Answer 5

Of the four angles, there are 2 pairs.
Since its a 4-sided polygon, the total of all angles is 360

Answer 6

Therefore, we know the acute angle also . It is ___

Answer 7

60?

Answer 8

Yes

Answer 9

Now, if you sketch the rhombus and draw the two diagonals, you'll see we have 4 triangles that are all the same (congruent).

Answer 10

See it?

Answer 11

yes, I do!

Answer 12

Now, these happen to be very special triangles.

Answer 13

They are 30-60-90 triangles.
When you have a moment, take the time to prove it, but not now OK?

Answer 14

I will :) since they are 30-60-90 does that mean we can find the area of the rhombus?

Answer 15

Sure, getting to that. In a 30-60-90 right triangle, the sides are in the ratio 1, sqrt(3), 2
with 2 on the hypotenuse, of course. OK?

Answer 16

Not sure if you're there. Now, for your triangles, we have to scale up, because we're given that the sides of the rhombus (hypotenuse of our triangles) is 6.
That means the sides of your triangles are actually 3 and 3 sqrt(3). Make sense?

Answer 17

yes it does!

Answer 18

Now, I think you can calculate the area.

Answer 19

thank you :)

Answer 20

One more thing, don't get mad, but there is an easier formula. Wanna know it?

Answer 21

yes please if you wouldn't mind!

Answer 22

Take the lengths of two adjacent sides and multiply them together, then times the sine of the included angle. Works for parallelograms too. You'll see it gives the same answer.

Answer 23

how would I find the sine of the included angle, would you mind explaining?

Answer 24

is that the number then hit sin on a scientific calculator?

Answer 25

We calculated the angle as 60 degrees. You can look it up in a table, or better yet, use the lengths for the 30-60-90 triangle I gave you above. sine = opposite over hypotenuse, which was sqrt(3)/2.

Answer 26

Or yes, you could do that :)

Answer 27

Just make sure its in degree mode.

Answer 28

Showing my age!

Answer 29

thank you so much for the help, it is really appreciated :)

Answer 30

yw