AREA OF PLANE FIGURES.
The area a of a rectangle, the lengths of two adjacent sides of which are b and c respectively, is a= bc. (X) The area a of the triangle ABC (Fig. 5) is fhb, (XI) a---ibc sin A, (XII) a s(s—a') (s—b) (s— c), (XIII) sin A•sin C or (XIV) sin (A +C)' where a', b, c are the lengths of the sides BC, CA, AB, respectively: h is the perpendicular distance (height) of B from AC; s is one-half of the sum of the sides: s= i (a' + b+ c).
If one angle of the triangle (say 4) is a right angle, both formulas (XI) and (XII) simplify: namely, the era is one-half the prod uct of the two sides which enclose the right angle (a =ibc). For a simple method of find ing the area of an equilateral triangle. (a'= b =c) multiply the square on one side by (a.----abc approximately), see formula (XVIII).
General Methods of Approximating the Length of Any Curved Line.— Divide the line into parts each of which differs but little from a small circular arc. Then each portion may be found by any one of the formulas (III), (IV) or (V). A convenient method for the
area a of the paral lelogram ABED (Fig. 5) is a =d (XV) Or a= cd sin D, (XVI) where c and d are the lengths of two adjacent sides, D is the angle between them, and h is the perpendicular distance between the two parallel sides whose length is d.
area a of the trapezoid CBED (Fig. 5), any four-sided figure with two of the opposite sides (BE and DC) par allel, is a •= ih(d+e), (XVII) where h is the perpendicular distance between the two parallel sides and d and e are the lengths of the parallel sides (dr—BE, e=DC).
Regular area a of a poly gon bounded by n equal sides each of length s and having its n angles all equal is n cot — a=s2X 4 (XVIII) The following table gives (approximately) the value of the multiplier in the simpler cases: A'B' is nearly coincident with its chord, the area a is a=APHAA'+PP'+QQ'+ . . . +TT')