122 The area of a rhombus formed on the hypothenuse of a right triangle, having the perpendicular of the triangle for its altitude, is a mean proportional between the areas of the hypothenuse and perpendicular. 15 15×15-225, square of the hypothenuse A B C D. 12×12-144, square of the perpendicular D E F G. 9×9 81, square of the base C G H I. 15×12=180, area of the rhombus D G L M. 180 is a mean proportional between 144 and 225. 12 6 H SECTION XVI. Semicircle. To find the area of a semicircle. Multiply the diameter by the radius, and the product by .7854. 1. What is the area of a semicircle, the diameter of which is 24? 24×12=288.7854-179.0712, the area. If the semicircumference only be given. Multiply the square of the semicircumference by .15916, the product will be the area. 2. If the diameter of a semicircle be 15 inches, what will be the area? ANS. 88.3575 square in. 3. What is the area of a semicircle, the diameter of which is 25 inches? ANS. 245.4375 sq. in. 4. What is the area of a piece of land in the form of a semicircle, the straight side of which is 40 feet? ANS. 628.32 sq. feet. 5. If a garden, in the form of a half circle, measures 18 feet from the centre of the straight side, to the centre of the curve, what number of square feet does it contain? ANS. 508.9392 sq. feet, SECTION XVII. Quadrant of a Circle. To find the area of a Quadrant of a Circle. What is the area of a quadrant of a circle, the radius of which is 12? 2 12X12=144X.7854-113.0976, the area. If the arc only be given: Multiply the square of the arc by .31832, the product will be the area. To find the length of an arc of a Circle. Multiply the diameter by .7854, the product is the length of a Quadrant. radius SECTION XVIII. Sectors of Circles. To find the area of the Sector of a Circle. Multiply the arc of the sector by half the radius. 1. which is 24 and radius 17? 24 17 Half of 17 is 8.5 and 24X8.5-204, the area. 2. What is the area of a sector, the arc of which is 58.68 and radius 17? 58. 68 Half of 17 is 8.5 and 58.68×8.5—498.78, the area. 3. If the radius of a sector be 12 inches and the arc be 3 feet, what is the area of the sector? ANS. 18 sq. ft. 4. What is the area of a sector of a circle, the diameter of which is 25 and arc 55 ? ANS. 687.5. 5. What is the area of a piece of land in the form of a sector of a circle, the curved side being 35 rods and the two straight sides being each 15 rods? ANS. 262 sq. rods. 6. What is the area of a sector of a circle, the arc of which is 45 degrees and radius 25 inches? ANS. 245.4375 sq. in. SECTION XIX. Segment of a Circle. To find the area of a Segment of a Circle. From the area of a sector of which the segment is a part, subtract the area of the triangle between the chord and radii; the remainder is the area of the segment. 24.024X5.25 126.126, area of the sector A B C D. 19.4X2 38.8, the area of the triangle A B C. 126.126-38.8-87.326, area of the segment A D C. |