help with triangles

May 23, 2006 12:41 PM

My room has an angled ceiling. I have measured its height at the highest and lowest points, and I have the dimensions of the room. Help me determine what height the ceiling is at a particular point on the floor, and where exactly the skylight is.

I remember the Pythagorean Theorem so I was able to compute the length of the ceiling. But I think it requires trigonometry that I don't remember to do the rest. What is the ceiling height above me if I am standing on the X, which is 66" from the wall?

Bonus question: the ceiling contains a skylight window. I can't reach it by a ladder, but I was able to approximate some measurements: from the window's low edge and from the top edge to the floor. With this information, can I now determine where exactly the window is between the highest and lowest points of the ceiling?

posted by xo to home & garden (11 comments total)

posted by xo at 12:42 PM on May 23, 2006

you don't actually need pythagoras. If you know the total length = L and the max height = H1 and the min height H2, the height at any distance X (lets call it Hx) measuerd from the min height point, is:

Hx = (H1-H2)*X/L
posted by signal at 12:58 PM on May 23, 2006

woops: that is ((H1-H2)*X/L)+H2
posted by signal at 12:58 PM on May 23, 2006

You could use trigonometry, but interpolation is quicker and simpler.

Let's say:
- A is the highest point, B is the lowest point.
- A is 20' high, B is 15' high.
- A and B are 25' apart.
- X is 5.5' (i.e. 66") from point B.

Then the height as X is:

(5.5' / 25' x (20' - 15') ) + 15' = 16.1'

Your bonus question isn't clear. Are you asking for the horizontal distance of the window from the wall, or something else?
posted by randomstriker at 12:58 PM on May 23, 2006

Ignore the rectangle at the bottom and focus only on the right triangle at the top. I think that for any right triangle, lines parallel to the bases form similar triangles (that is, the sides are proportional to each other) to the "main triangle," so it's just simple fractions from that point.

Assuming that your vertical 160" line is parallel to your 207" wall, then the 115-199-230 triangle is similar to the triangle formed by the vertical line extending upward from the X and the remainder of that triangle. That is:

115-199-230 corresponds to
X-133-Y

(X being the height from the base of your triangle to the roof, going upward from point X. Y being the irrelevant length of the roof (hypotenuse) of that triangle).

From that point:

115/X = 199/133

Simplifying 199/133 to 3/2:
115/X = 3/2
3X=230
X=~77
... add 92 for the height of the "rectangle" part of the room, and there you have it. I think.

You should be able to do something similar (ha) for the skylight measurements.
posted by Doofus Magoo at 12:58 PM on May 23, 2006

And "L" and "X" are both measured along the floor.
posted by signal at 12:59 PM on May 23, 2006

You can answer the first part with no trig at all using the law of similar triangles:

Use Y=199" - X = 133". In that case, H (of the triangle) is (133/199)* 115" = 78.86". Add the height of the rectangle (92") and you get: 168.86".
posted by JMOZ at 1:00 PM on May 23, 2006

As for your bonus question, I'm not sure if you want lengths along the floor or the ceiling, so I'll give you both (and how to calculate them!)

Length along the floor can be done using the same technique described above (similar triangles).

These distances are from the wall on the low-ceiling side:

199"*((146"-92")/115") = 93.44"
199"*((160"-92")/115") = 117.7"

On the ceiling, you'll want to use trig: First find the angle of the ceiling, for which you use theta= arctan(opposite/adjacent). = arctan (115"/199") = 30.02degrees.

Then, you use hypotoneuse=adjacent/Cos(theta) to get:
107.9" from the point of the triangle for the lower edge and 135.9" from the point of the triangle for the higher edge.
posted by JMOZ at 1:14 PM on May 23, 2006

If you are able to measure the height of the ceiling at its highest point, why don't you simply measure the height at the place where you, y'know, want to know how high it is?

Am I missing something here?
posted by dersins at 1:18 PM on May 23, 2006

dersins, there's a balcony along the high edge that enables me to measure from there to the ground, but at the other points in the ceiling away from my arms' length I can't reach.
posted by xo at 1:36 PM on May 23, 2006

One additional comment after rereading your tags- do keep in mind that actual construction is seldom (if ever) perfect, and even your ceiling might not be flat. If you want actual measurements with small tolerances, there is no substitute for actual measurements.

Also, keep in mind that my answer to the bonus (which I hope was clear) didn't account for the thickness of the skylight frame, etc. If you're trying to actually design something that will fit tightly somewhere, these numbers might not work.
posted by JMOZ at 3:23 PM on May 23, 2006