31. While the ordinary principles of mensuration are all that are necessary to calculate any roof area, yet the modern house, with its numerous gables and irregular surfaces, introduces complications which render some further explanation of roof measurement desirable. The most common error made in figuring roofs-and which 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.

32. The area of a plain gable roof, as shown in end and side elevations in Fig. 4, 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 the gable is found by multiplying the width of the gable ad by the altitude cb, and dividing by 2.

Fig. 4.

33. In Fig. 5 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 I m. In Fig. 6 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. 5. From the line a d, 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 a b, Fig. 5, and draw the lines ah and d f; then the figure a h f d will show the true shape and size of the roof shown in the elevation in Fig. 5. 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 c, the length of which is half the sum of the eave and deck lengths, while its breadth is the length of a common rafter.

Fig. 5.

34. A method of obtaining the lengths of valley rafters, applicable also to hip rafters, is shown in Fig. 7, 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 be the true length of the valley rafter a b.

Fig. 6.

Fig. 7.

35. As an example of roof mensuration, the number of square feet of surface on the roof shown in Fig. 8 will be calculated.

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 13.75 feet and 23 feet, respectively, and dividing by 2, the area is found to be 158.1 square feet. The area of the trapezoid g f i h is half the sum of f t 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 2/3 feet. Performing these operations, there results 5+14/2 X 10 2/3 = 101.3 square feet for each side, or 202.6 square feet for both. As the side gables are the same size, the area of the two roofs is 202.6x2 = 405.2 square feet.

Fig. 8.

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 158.1 square feet. The area of k n w is equal to half of n w, or 6.5 feet, multiplied by the true 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 7.5 feet. Then k n w = 6.5 x 7.5 = 48.7 square feet; whence q p n k equals 158.1 - 48.7 = 109.4 square feet.

The area of a p q c is ap + qc /2 multiplied by the true slope length of t v, or t v', which measures 15.25 feet. Substituting dimensions, the area is found to be 6+24 /2 X 15.25 = 228.7 square feet. From this deduct the area of y z u, which is the portion covered by the intersecting gable roof. The true length of t u along the slope is t u', measuring 12 feet; hence the area of y z u is 14 x 12 /2 = 84 square feet. The net area of a p q c is, therefore, 228.7 - 84 = 144.7 square feet; b c q w being equal to a p q c, its area is the same, making the area of both sides 289.4 square feet.

The area of k n m l is mn + lk/2 X ml' the slope length of m I. Substituting dimensions, the area is 11+16/2 X 8.5

= 114.7 square feet. As k l x w is equal to k n m l, the area of both is 229.4 square feet.

Adding the partial areas thus obtained, the sum is 158.1+405.2 + 109.4 + 289.4 +229.4 = 1,191.5 sq.ft.,or 11.9 squares.