Question

Prove the area of a parallelogram is 1/2 d_1 d_2 sin(theta).

Answer 1

Is there a figure? If d_1 and d_2 are the side lengths, this isn't quite right

Answer 2

They are the diagonals.

Answer 3

Can you draw a figure, or shall I?

Answer 4

http://upload.wikimedia.org/wikipedia/commons/thumb/4/41/Parallelogram.svg/220px-Parallelogram.svg.png

Answer 5

And what is theta?

Answer 6

E

Answer 7

While you're thinking, I'll point out that the figure already has the diagonal segments labeled to show that at the center, each diagonal is divided in half. You can prove that easily from the fact that the sides are parallel. So the area of triangle DAC is 1/2 the total.

Answer 8

You've got me stumped for the moment, better ask for help.

Answer 9

My main issue is: what does sin(theta) give me, the height? The base?

Answer 10

Right. I think of rearranging the pieces, and get lost. I think of direct calculation and it gets to be a mess. I think of the law of sines, and hope for cancellations. That'd be my bet.

Answer 11

How about this
Let d_1 = the diagonal with a single mark, and d_2 the other one.
Let b = the base of the parallelogram (DC) and s the side (BC).
Let theta = the angle EDC.
Now
sin theta / (1/2) d_2 = sin E / b

Answer 12

And the area of BDC = d_1 sin theta times b

Answer 13

Of a triangle, the area of .5 a b sin(c). .5 of diagonal one and 2 * sin (E) gives me one of the triangle areas. sin(180-E)=sin(E), so we can find the area of the other triangle, and multiply the sum of those two triangles by two.

Answer 14

... to get the total area.

Answer 15

It's sounding good.

Answer 16

I'm having a problem of a factor of 1/2. Doing it my way:

sin theta / (d_2/2) = sin E / b
sin theta = (d_2/2) (1/b) sin E

Area of DBC (half the whole) = d_1 b sin theta
A (1/d_1) (1/b) = sin theta = (d_2/2) (1/b) sin E
cancel (1/b)
A = d_1 (d_2/2) sin E

But that's only half the area.
total = d_1 d_2 sin E
What do you think?

Answer 17

Wanted to show another way. I cut the solid into 4 pieces and rearranged them.
See the figure.
Standard result with vectors is that the area is A x B which here is d_1 d_2/2 sin E
http://en.wikipedia.org/wiki/Cross_product

Answer 18

Use the second figure, first one's bad.
Was fun!