Calculating the Angle of a Plank

Date: 08/22/2001 at 09:36:50
From: nick pounder
Subject: Calculating the angle of a plank.

I've been through all the weird and wonderful high school trig 
formulas, tried all sorts of substitutions and simultaneous equations, 
but I have not been able to crack this. On the face of it it looks 
very simple and is a very practical problem: 

   a|.
    |   .
   b|.     .
    |   .     .
    |      .     .
    |_________.___ .
   0          c    d

A hypothetical plank bounded by the dots is leaning against a wall. 
All I want to know is the angle the plank makes.  The problem is: I 
know the height a and I also know the length c, the first point 
of contact with the floor. However, I don't know length d, or height b 
for that matter. 

I do know that the plank is z width. If I knew the length d I could 
calculate the angle easily; however, I don't know how much farther on 
from c that d is, because I don't know the angle.  

Are there any kind of simultaneous trig equations that could be used 
to solve this for a plank of known width? And if the solution is 
embarassingly simple, apologies in advance!   

The diagram above isn't very good as it shows the plank already cut to 
the correct angle; in practice my new plank has 90-degree corners.  I 
want to calculate the angle without further measurements, so I know 
what angle to cut the plank so it rests smoothly against both the wall 
and floor.

Thanks very much!

Date: 08/23/2001 at 10:27:43
From: Doctor Rick
Subject: Re: calculating the angle of a plank.

Hi, Nick, thanks for writing to Ask Dr. Math.

This is not a trivial problem, no apologies are needed! 

Let's solve the problem using coordinate geometry; you have already 
set up your figure this way. I'll add a few lines to your figure:

        a *-
         /|  -
        / |    -
      z/th|       -
      /   |          -
(x0, *    |            -
 y0)   -  |               -
         -|                 -
        b *-                   -
          |  -                    -
          |     -                   -
          |       -                    -
          |          -                   -
          |             -                   -
          |               -                   -
          |                  -                   -
          |                     -                   -
          |                        -                  -
          |                          -                   -
          |                             -                  -
          |                                -                  -
          |                           theta  -           theta   -
          +-------------------------------------*------------------*
         0                                      c                  d

You know a, c, and z. In terms of the unknown angle theta, we can find 
the coordinates of the point (x0,y0) that I added to the figure:

  x0 = -z*sin(theta)
  y0 = a - z*cos(theta)

the slope of the line marking the bottom edge of the plank is

  m = -tan(theta)

Thus the equation of this line is

  y = m(x - x0) + y0

  y = -tan(theta)(x + z*sin(theta)) + a - z*cos(theta)

This line passes through the point (c,0), so

  0 = -tan(theta)(c + z*sin(theta)) + a - z*cos(theta)

We want to solve this equation for theta. To do this, we rewrite 
tan(theta) and cos(theta) in terms of sin(theta):

  cos(theta) = sqrt(1 - sin^2(theta))

  tan(theta) = sin(theta)/cos(theta)
             = sin(theta)/sqrt(1 - sin^2(theta))

Call s = sin(theta) to simplify the equations:

  cos(theta) = sqrt(1 - s^2)
  tan(theta) = s/sqrt(1 - s^2)

  0 = -s/sqrt(1 - s^2)(c + zs) + a - z*sqrt(1 - s^2)

Multiply through by sqrt(1 - s^2):

  0 = -s(c + zs) + a*sqrt(1 - s^2) - z(1 - s^2)

Simplifying, the s^2 term cancels out:

  sc + z = a*sqrt(1 - s^2)

Square both sides (we'll need to check that we don't introduce 
spurious solutions at this step):

  (sc + z)^2 = a^2(1 - s^2)

  (a^2+c^2)s^2 + 2zcs + (z^2-a^2) = 0

Solve this quadratic equation for s:

  s = (-2zc +or- sqrt(4z^2c^2-4(a^2+c^2)(z^2-a^2)))/(2(a^2+c^2))
    = (-zc +or- sqrt(z^2c^2-(a^2+c^2)(z^2-a^2)))/(a^2+c^2)
    = (-zc +or- sqrt(a^4+a^2c^2-a^2z^2))/(a^2+c^2)
    = (-zc +or- a*sqrt(a^2+c^2-z^2))/(a^2+c^2)

The negative solution is spurious, since we know that theta is in the 
first quadrant, so sin(theta) is between 0 and 1. Thus, setting 
sin(theta) equal to s, we have our solution:

+----------------------------------------------------+
| theta = sin^-1((a*sqrt(a^2+c^2-z^2)-zc)/(a^2+c^2)) |
+----------------------------------------------------+

Let's check the formula by plugging in the numbers for a simple case. 
If z = 1 and theta = 45 degrees, then we can see from the figure that 
a = b+sqrt(2) and c = b. Make these substitutions:

  sin(theta) = ((b+sqrt(2))*sqrt((b+sqrt(2))^2+b^2-1)-b)
                /((b+sqrt(2))^2+b^2)

             = ((b+sqrt(2))*sqrt(2b^2+2b*sqrt(2)+1)-b)/(2b^2+2b*
                  sqrt(2)+2)

             = ((b+sqrt(2))*(sqrt(2)*b+1)-b)/(2b^2+2b*sqrt(2)+2)

             = (b^2*sqrt(2)+2b+sqrt(2))/(2b^2+2b*sqrt(2)+2)

             = sqrt(2)(b^2+b*sqrt(2)+1)/(2(b^2+b*sqrt(2)+1)

             = sqrt(2)/2

This is indeed the sin of 45 degrees, so the formula checks out in 
this case.

I hope this is helpful to you. I'm assuming you can finish the work, 
finding the length of at least one edge of the plank so you can cut it 
to fit.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/