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### The Area of a Roof

```Date: 05/18/2002 at 20:07:13
From: Earline M. Simms
Subject: Determing the area of a roof and materials neded

1.  How do I determine the area of a hipped roof on a building that is
100 feet long and 80 feet wide? The pitch of the roof is 6/12, and it
has a 3 foot overhang.

2. After I know the area of the roof, how do I determine the number of
sheets of plywood, the number of rolls of felt, and the number of
bundles of shingles needed to cover the roof?

I don't know how to get started on solving this problem.

Thank you very much for your help.
```

```
Date: 05/19/2002 at 16:51:25
From: Doctor Rick
Subject: Re: Determing the area of a roof and materials neded

Hi, Earline.

I am not too knowledgeable about construction, and we are focused on
helping high-schoolers and below, so I will just show the geometry
involved in your questions and leave the rest to you.

The plan view (from above) looks like this, assuming the overhang
means that the eaves are 3 feet horizontally out from each wall:

+--------------------------------------------------+
| \                                              / |
|   \                                          /   |
|     \                                      /     |
|       \                                  /       |
|         \                              /         |
|           \                          /           |
|             \                      /             |
|               \                  /               |
|                 \              /                 |
|...................\__________/                   |86 ft
|       43 ft       /:  20 ft  \                   |
|                 /  :           \                 |
|               /    :             \               |
|             /      :               \             |
|           /        :                 \           |
|         /          :43 ft              \         |
|       /            :                     \       |
|     /              :                       \     |
|   /                :                         \   |
| /                  :                           \ |
+--------------------------------------------------+
106 ft

The length of the peak is 106 feet minus twice the horizontal
distance from the eaves:

106 - 2 * 43 = 106 - 86 = 20 feet

If I understand the pitch terminology correctly, 6/12 means that for
every 12 feet horizontally, the roof rises 6 feet. Thus in 43 feet
the roof rises

43 ft * 6/12 = 21.5 feet

This is the height of the peak over the eaves. Looking from the side,
we see a triangle like this:

+
/  :  \
/     :     \
/        :        \
/           :21.5 ft    \
/              :              \
/                 :                 \
/____________________:____________________\
43 ft               43 ft

We can find the length of the slope using the Pythagorean theorem:
the hypotenuse (long side, opposite the right angle) of a right
triangle is the square root of the sum of the squares of the other
two sides.

sqrt(43^2 + 21.5^2) = sqrt(1849 + 462.25)

= sqrt(2311.25)

= 48.075 feet

This distance is the height of both the triangular and the
trapezoidal roof sections. We can find the areas of these sections
using formulas found in the Dr. Math FAQ under "Formulas".

Area of triangle = Base * Height / 2

= 86 ft * 48.075 ft / 2

Area of trapezoid = (Base1 + Base2) * Height / 2

= (106 ft + 20 ft) * 48.075 ft / 2

The total roof area is twice the sum of these, since the roof
consists of two of each type of section.

You can figure the number of sheets of plywood, rolls of felt and
squares of shingles by dividing the roof area by the area of a sheet
of plywood, a roll of felt, or a square of shingles. In some of these
cases, though, there will likely be considerable waste, so I wouldn't
count on needing only what this calculation indicates.

I hope this helps!

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Higher-Dimensional Geometry
High School Triangles and Other Polygons

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