Building a Manger

Date: 12/03/2001 at 22:39:44
From: Anne Markle
Subject: Angles for a building

I am building a manger with a base of 11" (A), one wall 7 1/2" high 
(B) and the other wall 6" high (C). Both walls meet the base at 
90-degree angles. My question is:

What would the length of my roof be and what would be the two angles 
of the walls meeting the roof?

Thanks.

Date: 12/04/2001 at 01:08:51
From: Doctor Jeremiah
Subject: Re: Angles for a building

Hi Anne,

Many people write to us asking how math fits into the real world.  I 
see that isn't your problem, though!


   +
   |    +
   |         +
   |              +
   |                   +
   |                   |
  7 1/2                6
   |                   |
   |                   |
   +--------11---------+

This is a side view of your building. The roof must go on top, but 
first you need to know how long it will be.

Lets redraw this as a rectangle and a triangle:

   +
   |    +
  1 1/2      +
   |              +
   +--------11---------+
   |                   |
   6                   6
   |                   |
   |                   |
   +--------11---------+

You will notice that the angle where the 11" top of the rectangle 
meets the 1 1/2" extension is 90 degrees. Have you heard of 
Pythagoras? He is famous for his method of calculating the third side 
of a triangle that has a 90 degree angle, the Pythagorean theorem.

Pythagoras said that the square of the length of the long side was 
equal to the sum of the squares of the other sides.

So if we call the length of the roof "R" then because of Pythagoras we 
can say:

  R squared = 1.5 squared + 11 squared
  R squared = 1.5 times 1.5 + 11 times 11
  R squared = 2.25 + 121
  R squared = 123.25

And if R squared is 123.25 then R is the square root of 123.25, which 
is 11.1 inches.

But wait! Roofs usually hang over the edge, so you are going to want 
to make it longer than that anyway. I would suggest 12 inches so that 
you get a decent size for the eaves.

As for the angles, they require some trigonometry to calculate. It 
turns out that we don't need to know the length of the third side to 
calculate the angles.

  a
   +
   |    +
  1 1/2      +
   |              +
   +--------11---------+ b

I have labeled the two angles a and b. The angle at  a  can be 
calculated by taking the tangent of 11/1.5, and the angle at  b  can 
be calculated by taking the tangent of 1.5/11 - but you will need a
calculator to do that.

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