A plane figure bounded by more than four lines is called a polygon. It must therefore have at least five sides, and the number of sides which it may have is not limited. In this work will be introduced only the forms in common use, for the purpose of showing simple methods of estimating their areas A regular polygon has all its sides and angles equal, as shown in Fig. 36. An irregular polygon has its sides and angles unequal, as shown in Fig. 37.
Fig. 36. - A Regular Polygon.
Fig. 37. - An Irregular Polygon.
A polygon of five sides, as shown in Fig. 36 or 37, is called a pentagon. The diagonal is a straight line drawn between any two angular points of a polygon. The diameter is a straight line drawn from any angle through the center to the opposite side or angle, as the case may be.
To find the area of a regular pentagon we will let A B C D E represent the sides of a regular pentagon, as shown in Fig. 38. Draw the diameter A F and connect E with B, which divides the pentagon into four figures - namely, two right angled triangles of equal areas and two trapezoids of equal areas. E G multiplied by G A will give the area of the two triangles. Half the sum of D C and E B multiplied by G F will give the area of the two trapezoids. The two areas added will give the total area.
Fig. 38. - Finding Area of Regular Pentagon.
Fig. 39. - Finding Area of an Irregular Pentagon.
To find the area of an irregular pentagon, we will let A B C D E represent the sides, as shown in Fig. 39. Next draw A D and A C, which will divide the pentagon into three triangles of unequal areas; then draw the altitude of these triangles, which is the perpendicular distance from their vertices to the opposite sides, called the base and shown by the lines E F, A G and B H. This divides the figure into six right angled triangles of unequal areas. A D multiplied by half the altitude E F will give the area of triangles 1 and 2, or A E D ; then D C multiplied by half the altitude A G will give the area of triangles 3 and 4, or D A C. Again A C multiplied by half the altitude H B will give the area of triangles 5 and 6, or A B C. The three areas added will give the total area.
Fig. 40. - A Hexagon.
Fig. 41. - Finding the Area of a Hexagon.
Fig. 42. - Describing any Regular Polygon.
Fig. 43. - An Octagon.
A polygon of six sides is called a hexagon, and is shown in Fig. 40. To find the area of this figure draw the diagonals as shown in Fig. 41, which divide the hexagon into equal triangles, the size of which is represented by A B C. Next draw the altitude of this triangle, as shown by the dotted line B D. Now, A C multiplied by half the altitude B D will give the area of the triangle A B C, and this mul tiplied by six will give the total area. The area of any regular polygon may be found by drawing lines from all of its angles to the center, thus forming triangles of equal areas, which may be estimated by multiplying the base by one-half the altitude, as shown in Fig. 41. To describe any regular polygon draw the circumference of a circle; divide the circumference into as many equal spaces as the polygon has sides, connect these points with straight lines, and the polygon is completed, as shown in Fig. 42.
Fig. 44. - Plan of an Octagon Tower Roof.
Fig. 45. - An Elevation of an Octagon Tower Hoof.
A polygon of eight sides is called an octagon and is shown in Fig. 43. In Fig. 44 is represented a plan and in Fig. 45 an elevation of an octagon tower roof. In Fig. 45 A B C D represent the plates and A E, B E, C E and D E the hip rafters. The dotted line F E represents the common rafter. To find the area of this roof multiply B C by half of F E and this product by eight, the number of sides. It will now be seen that the area of any tower roof from a square to a polygon of any number of sides may be found by multiplying the length of its side by half the length of the common rafter. If the tower has a round base then the circumference of its base multiplied by half the length of the common rafter will give the area. The reader has now been shown wherein it is possible to make mistakes in the measurement of roofs, as indicated by the elevations. It has been shown how to develop the true shapes and sizes of irregular roof surfaces and how to reduce them to squares or rectangles of equal areas, or to figures whose areas are easily calculated. I might go on illustrating and describing roofs seemingly without end, but enough has been illustrated to thoroughly show the principles and methods of estimating roof surfaces. By a little study of the principles and methods, as previously set forth, the reader will be able to make proper application of them to the surface measurement of any roof.
It will be noticed in nearly all cases that the essential measurements for computing the area or surfaces of roofs are - 1, the length at the eaves ; 2, the length at the ridge or deck, as the case may be, and 3, the length of the common rafter.
In works of this kind it has. been customary to show a number of illustrations on geometry, merely indicating how to construct certain figures from a given side or a few given points, while in all cases the most important part which a carpenter requires - that of computing the area of irregular surfaces - has been omitted. In the art of carpentry there is no place in which these irregular-shaped figures appear as frequently as they do in the construction of roofs, and if the carpenter has no accurate methods for computing their areas then he has to make a guess, which is the course taken by many who have never seen a proper application of geometry to the surface measurement of roofs. Roof surfaces have to be estimated in order to ascertain the amount of material required to cover them, as the sheeting, shingles, slate, tin, copper, iron, etc, or whatever may be used for the roof covering. In the illustrations and examples given there might have been presented many rules for finding the length of certain sides of a figure, by having the lengths of one or more of the other sides, but they would be merely mathematical problems, which in most cases could be solved only by square root. As many carpenters, are not conversant with square root it has been deemed best to avoid its use as much as possible in this work, and especially in places where it is not needed. It must be generally conceded in taking roof measurements, that if a carpenter can measure one distance he can measure the roof to find any distance he may desire to know. Therefore the illustrations given have been more to show how to measure roofs to obtain the proper dimensions for computing their areas than as geometrical problems and methods of construction. The author has considered the subject of roof measurement worthy a place by itself in estimating, and the subject of roof framing will be taken up, thoroughly illustrated and described in another part of this work.