# Finding the dimensions of a rotated rectangle inside another rectangle.

1. May 2, 2009

### galneweinhaw

1. The problem statement, all variables and given/known data
If I have a rectangle rotated at a known angle with respect to a rectangle of known dimensions that inscribes it, how can I find the dimensions of the inscribed/inner rectangle?

http://bp3.blogger.com/_4Z2DKqKRYUc/Rnz_BgODzFI/AAAAAAAAAIw/uj_cVfPI8D4/s1600-h/Img_6-23-07_Blog.jpg

If the image above is my example, I know the dimensions of ABCD and I know all the angles, such as < BPQ.

2. Relevant equations
Trig/Pythagorous...

3. The attempt at a solution
I'll post if I come up with anything that looks like it's gettign anywhere =P

Thanks for the help... let's see how my first ever post is received =)

2. May 2, 2009

### diazona

Surely you must have tried something?

Hint: can you find the four triangles in the figure? From there, you have the trig formulas to calculate the lengths of the sides you need...

3. May 2, 2009

### galneweinhaw

oh, I've been trying for a couple hours. But I haven't really made it anywhere =(

4. May 2, 2009

### Staff: Mentor

So show us what you've tried.

5. May 2, 2009

### galneweinhaw

ok... I think I have something that should be able to go somewhere...

Here's a relabelled image:

ɵ, X, and Y are known, trying to find h and w.

y1, y2, x1, x2, w, and h are the unknowns (6)

I can get seven equations:

w2 = x22+y12

h2 = x12+y22

Y = y1 + y2

X = x1 + x2

y1 = x2 tanɵ

x1 = y2 tanɵ

XY = x2y1 + x1y2 + hw (areas)

6. May 2, 2009

### galneweinhaw

eliminating x1,x2,y1,y2 I get...

XY = $$\frac{(w^2+h^2)tan\theta}{1+tan^2\theta} + hw$$

X = $$\frac{htan\theta + w}{\sqrt{1+tan^2\theta}}$$

Y = $$\frac{wtan\theta + h}{\sqrt{1+tan^2\theta}}$$

edit:
sub some trig identities

XY = $$(w^2+h^2)sin\theta cos\theta + hw$$

X = $$(htan\theta + w)cos\theta$$

Y = $$(wtan\theta + h)cos\theta$$

Last edited: May 2, 2009
7. May 2, 2009

### galneweinhaw

Further simplifying...

$$X = hsin\theta + wcos\theta$$

$$Y = wsin\theta + hcos\theta$$

LOL.... I could have pulled that directly off the diagram! well, at tleast I know my algebra is sound =P

8. May 10, 2011

### igtech

But with this, you find X and Y that it is supposed you already knew, what about finding h and w , huh??

9. Jun 22, 2011