Chap. VIII.
GEOMETRY.
873
Fig. 1134.
Fig. 113.*).
To measure a regular Polygon. — Add all the sidestogether, and multiply the sum by a perpendicular let fallfrom the centre to one side ; half the product will be thearea.
To find the superficial content of the pentagon ABCDE,measure one side DC, which is 22 feet, and then multiplyit by 5, the number of the sides : multiply its product 110bv 15, the height of the perpendicular F G ; take half thisproduct of 1650, which is 825 superficial feet, for the area.
"When the figure is inaccessible, surround it by the squareor rectangle FGIII, whose side FG is 35 feet 3 inches, andFI 33 feet 6 inches 4 parts ; when these two sides are mul-tiplied together, their product will be the area of the rect-angle : then measure the four triangles FAE, AGB,
E D I, and BHC; when their areas are found and addedtogether, deduct the sum from that already found for thewhole rectangle, and the remainder will be the area of thepentagon. Similar polygons inscribed in circles have, as wehave already seen, their perimeter in the same ratio as thediameters of those circles ; for if we imagine the number ofsides to be infinite, or so numerous that they coincide withthe circumference of the circle, then the perimeter of sucha polygon may be regarded the same as the circumferenceof the circle.
To measure an irregular Poly-gon. — Plant piquets at all theangles of the irregular polygon,and one in the centre of thefigure, in order that cords maybe stretched from thence to theangles ; by this means the ir-regular figure will be dividedinto triangles, which may bemeasured separately, and thewhole added together.
To measure the irregular hexagon ABCDEF, plant piquets at all the angles, and onein the centre; then stretch cords to the several angles, and when they are defined thelengths of their sides may be obtained, and their areas found, which added together will bethat of the whole.
If the hexagon ABCDEFhas a piece of water in the cen-tre, and is inaccessible, thencords may be stretched from op-posite angles, and by this meansmeasured, and the area found.
If the entire area is inaccessible,then the whole figure may bebounded by a rectangle, and allthe outer angles measured.
In measuring land, take asfew dimensions as possible;measure, for instance, FC, and dropperpendiculars from A, B, D, E,and measure them. If thefigures are very irregular, asGHIKLMNOPQRSTVX,form it into trapeziums andangles.
It must always be borne inmind that a vast number of smallfigures tend to confuse the sur-veyor and render his calculationsmore difficult; it is therefore ad-visable not to have a greaternumber of triangles than is abso-lutely necessary, for in findingtheir area it is impossible to avoid
Fig. 1137.
Fig. 1138.
Fig. 1139.