CHAPTER III. MEASUREMENT OF SURFACES. 805. By 248, any parallelogram is equivalent to the rectangle of its base and altitude; therefore, 806. COROLLARY. The area of a parallelogram divided by the base gives the altitude. 807. By 252, any triangle is equivalent to one-half the rectangle of its base and altitude; therefore, Given, one side and the perpendicular upon it from the opposite vertex, to find the area of a triangle. Take half the product of the base into the altitude. RULE. A = Jab. 808. Given, the three sides, to find the area of a triangle. RULE. From half the sum of the three sides subtract each side separately; multiply together the half-sum and the three remainders. The square root of this product is the area. FORMULA. - a) (sb) (s — c). PROOF. Calling j the projection of c on b, by 306, 809. To find the area of a regular polygon. RULE. Take half the product of its perimeter by the radius of the inscribed circle. PROOF. Sects from the center to the vertices divide the polygon into congruent isosceles triangles whose altitude is the radius of the inscribed circle, and the sum of whose bases is the perimeter of the polygon. 810. To find the area of a circle. Ө RULE. Multiply its squared radius by τ. FORMULA. 0 = r2π. If a regular polygon be circumscribed about the circle, its area N, by 809, is arp. If, now, the number of sides of the regular polygon be continually doubled, the perimeter decreases toward c as limit, and N toward circle. But the variables N and are always in the constant ratio r; therefore, by 798, Principle of Limits, their limits are in the same ratio, RULE. Multiply the length of the arc by half the radius. S = S = {lr = {ur2. PROOF. By 506, S: 0 ::/: c :: u : 2π, 812. An Annulus is the figure included between two concentric circles. Its height is the difference between the radii. 813. To find the area of a sector of an annulus. RULE. Multiply the sum of the bounding arcs by half the difference of their radii. - · FORMULA. S. A . = } (r2 − r1) (4, + 4) = zh (4, + 4). PROOF. The annular sector is the difference between the two sectors r,, and fr.l.. But, and, are arcs subtending the same angle; therefore, by 804, |