=64.= While a knowledge of how to apply the ordinary principles of
mensuration is all that is necessary to calculate any roof area, yet
the modern house, with its numerous gables and irregular surfaces,
introduces complications that render some further explanation of roof
measurement desirable. The most common error made in figuring roofs—and
one that should be carefully guarded against—is that of using the
apparent length of slopes, as shown by the plan or side elevations,
instead of the true length, as obtained from the end elevations.
[Illustration: FIG. 6]
=65.= The area of a plain gable roof, as shown in end and side
elevations in Fig. 6, is found by multiplying the length _g j_ by the
slope length _b d_, and further multiplying by 2, for both sides. The
area of each gable is found by multiplying the width of the gable _a d_
by the altitude _c b_, and dividing by 2.
[Illustration: FIG. 7]
[Illustration: FIG. 8]
[Illustration: FIG. 9]
=66.= In Fig. 7 is shown the plan and elevation of a hip roof, having a
deck _z_. The pitch of the roof being the same on each side, the line
_c d_ shows the true length of the common rafter _l m_.
In Fig. 8 is shown the method of developing the true lengths of the
hips and the true size of one side of the roof. Let _a b c d_ represent
the same lines as the corresponding ones in Fig. 7. From the line _a
d_, Fig. 8, through _b_ and _c_, draw perpendiculars, as _g h_ and _e
f_; lay off from _g_ and _e_ on these lines, the length of the common
rafter _c d_, Fig. 7, and draw the lines _a h_ and _d f_, Fig. 8; then
the figure _a h f d_ will represent the true shape and size of the side
of the roof shown in the elevation in Fig. 7. The area of the triangle
_d e f_ is equal to the area of the triangle _a g h_ or a similar
triangle _a i h_. Hence, the portion of the roof _a h f d_ is equal in
area to the rectangle _a i f e_, the length of which is half the sum of
the eave and deck lengths, while its breadth is the length of a common
rafter.
=67.= A method of obtaining the lengths of valley rafters, applicable
also to hip rafters, is shown in Fig. 9, which is the plan of a
hip-and-gable roof. To ascertain the length of the valley rafter _a b_,
draw the line _a c_ perpendicular to _a b_ and equal in length to the
altitude of the gable; then draw the line _c b_, which will represent
the true length of the valley rafter _a b_.
=68.= As an example of roof mensuration, the number of square feet of
surface on the roof shown in Fig. 10 will be calculated.
[Illustration: FIG. 10]
The area of the triangular portion _a c b_ is equal to the slope length
of _d c_ (found by laying off _c′ c_ equal to the height of the ridge
above the eaves and drawing _c′ d_) multiplied by the length of the
eaves line _a b_ and divided by 2. Multiplying the dimensions 13.5 feet
and 23 feet, respectively, and dividing by 2, the area is found to be
155.3 square feet.
The area of the trapezoid _g f i h_ is half the sum of _f i_ and _g h_
(shown in their true length on the plan) multiplied by the true length
of _h i_. The latter is found by marking the height of the gable _i i′_
on the ridge line, and drawing the line _i′ h_, which measures 10.6
feet. Performing these operations, there results
5 + 14
------ × 10.6 = 100.7 square feet
2
for each side, or 201.4 square feet for both. As each of the side
gables is the same size, the area of the two roofs is 201.4 × 2 = 402.8
square feet.
The area of the polygon _q p n k_ is equal to the triangle _q p w_
minus the triangle _k n w_, the area covered by the intersecting gable
roof. The former is equal to the triangle _a c b_, the area of which
is 155.3 square feet. The area of _k n w_ is equal to half of _n w_,
or 6.5 feet, multiplied by the true length of _k s_ or the altitude of
the triangle; the latter is obtained by laying off _k k′_ equal to the
height of the gable, 5.5 feet, at right angles to _k s_, and drawing _s
k′_, which is the required altitude and which measures almost 7.4 feet.
Then _k n w_ = 6.5 × 7.4 = 48.1 square feet; whence _q p n k_ equals
155.3 - 48.1 = 107.2 square feet.