The octagon may be drawn in a square and its area will be that of the square, less the four triangles in the corners. ([Fig. 237]). So the problem resolves itself into finding the area of one of these triangles. If we knew the length of one of the sides of the octagon, the solution would be simple, but we only know that the eight sides are equal. The following method may be worked out: Find the diagonal of the sixteen foot square. It is 22.6+. Deduct the distance across the flats, 16, leaving 6.6 feet equally divided between a and b; a = 3.3 and it may be proved that c = a = d. So in each corner we have a triangle whose base is 6.6 × 3.3. The area of a triangle equals half its base by the altitude. Therefore the area of each triangle is 3.3 × 3.3 and 3.3 × 3.3 × 4 equals 43.56 square feet, the combined area of the four corners. This deducted from the area of the square leaves the area of the octagon, or 256-43.56 = 212.44 square feet.
Fig. 238. Problem of the hexagon
Assume that our problem is to find the narrowest board we can use to cut out a hexagon whose diameter is fourteen inches. As shown in Chapter IV, the hexagon is drawn in a circle. One of the sides is equal to the radius or half the diameter. This gives us the arrangement shown in [Fig. 238], in which our problem is confined to the right-angled triangle whose base is seven and hypothenuse fourteen. From our knowledge of triangles, we deduct the square of seven (49) from the square of 14 (196), leaving 196-49 = 147, which is the square of the altitude. Then √147 = 12.12, which is the narrowest board from which we can obtain a hexagon 14 inches in diameter.
These examples are given to show the close connection between woodwork and arithmetic.
[LII]
LUMBER: NO. 1
It is hardly possible for a boy to select and purchase wood for his various purposes without some knowledge of the different woods and their peculiar characteristics. No two are exactly alike, and in fact two trees of the same kind growing in different parts of the country under different conditions will produce timber of very different qualities. This is specially noticeable in the tulip or white wood, for example. A tree of this species, growing in a swamp in the South, will yield a very different wood from one grown on high ground in the North.