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

• galneweinhaw
In summary, the author has attempted to find the dimensions of an inscribed/inner rectangle by solving equations. They say that they can get seven equations, but that they do not know what h and w are.

## Homework Statement

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?

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

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.

## Homework Equations

Trig/Pythagorous...

## 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 =)

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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...

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

So show us what you've tried.

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

Here's a relabelled image:
http://img7.imageshack.us/img7/764/rectb.jpg [Broken]

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 + homework (areas)

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

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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

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

## 1. What is a rotated rectangle?

A rotated rectangle is a rectangle that has been rotated around a certain point, changing its orientation from its original position.

## 2. Why would I need to find the dimensions of a rotated rectangle inside another rectangle?

Finding the dimensions of a rotated rectangle inside another rectangle can be useful in various scenarios, such as designing buildings, creating computer graphics, or solving geometry problems.

## 3. How do I find the dimensions of a rotated rectangle inside another rectangle?

To find the dimensions of a rotated rectangle inside another rectangle, you will need to know the measurements of the original rectangle, the angle of rotation, and the coordinates of the rotated rectangle's corners. You can then use mathematical formulas to calculate the dimensions.

## 4. Can I use a calculator to find the dimensions of a rotated rectangle inside another rectangle?

Yes, you can use a calculator to find the dimensions of a rotated rectangle inside another rectangle. However, make sure to use the correct formulas and enter the measurements accurately for accurate results.

## 5. Are there any online tools or resources that can help me find the dimensions of a rotated rectangle inside another rectangle?

Yes, there are various online tools and resources available that can help you find the dimensions of a rotated rectangle inside another rectangle. These include calculators, step-by-step guides, and video tutorials. Make sure to use trusted and reliable sources for accurate results.