3d. If the segment is greater than the semicircle, add the two areas together; but if it is less, subtract them, and the result in either case will be the area required. and, = AD = √AP2+ PD = √144 + 16 = 12.64911: 16 = 4, Then, area of sector AFBC = 279.36 do. of triangle ABC = 56.6825 gives area of segment AFB = 336.0425 3. What is the area of a segment, the radius of the circle being 10, and the chord of the arc 12 yards? Ans. 16.324 sq. yds. 4. Required the area of the segment of a circle whose chord is 16, and the diameter of the circle 20. Ans. 44.5903. 5. What is the area of a segment whose arc is a quadrant-the diameter of the circle being 18? Ans. 63.6174. 6. The diameter of a circle is 100, and the chord of the segment 60: what is the area of the segment? Ans. 408, nearly. 65. How do you find the area of a circular ring; that is, the area included between the circumferences of two circles, having a common centre? 1st. Square the diameter of each ring, and subtract the square of the less from that of the greater. 2d. Multiply the difference of the squares by the decimal .7854, and the product will be the area. 2. What is the area of the ring when the diameters of the circles are 20 and 10? Ans. 235.62. 3. If the diameters are 20 and 15, what will be the area included between the circumferences? Ans. 137.445. 4. If the diameters are 16 and 10, what will be the area included between the circumferences? Ans. 122.5224. 5. Two diameters are 21.75 and 9.5; required the area of the circular ring. Ans. 300.6609. 6. If the two diameters are 4 and 6, what is the area of the ring? Ans. 15.708. 66. How do you find the area of an ellipse? Multiply the two axes together, and their product by the decimal .7854, and the result will be the required area. 2. Required the area of an ellipse whose axes are 24 and 18. Ans. 339.2928. 3. What is the area of an ellipse whose axes are 35 and 25? Ans. 687.225. 4. What is the area of an ellipse whose axes are 80 and 60? Ans. 3769.92. 5. What is the area of an ellipse whose axes are 50 and 45? SECTION II. Ans. 1767.15. MENSURATION OF SOLIDS. 1. What is a solid or body? A solid or body is that which has length, breadth, and thickness. 2. What is a body bounded by planes called? Every solid bounded by planes is called a polyedron. 3. What are the bounding planes, the straight lines, and the angular points called? The planes which bound a polyedron are called faces. The straight lines in which the faces intersect each other, are called the edges of the polyedron; and the points at which the edges intersect, are called the vertices of the angles, or vertices of the polyedron. 4. What is a prism? What are its bases? what its convex surface? A prism is a solid, whose ends are equal polygons, and whose side faces are parallelograms. Thus, the prism whose lower base is the pentagon ABCDE, terminates in an equal and parallel pentagon FGHIK, which is called the upper base. The side faces of the prism are the parallelograms DH, DK, EF, AG, BH. These are called the convex or lateral surface of the prism. 5. What is the altitude of a prism? K. F I H E D B C A The altitude of a prism is the distance between its upper and lower bases; that is, it is a line drawn from a point of the upper base, perpendicular to the lower base. 6. What is a right prism? A right prism is one in which the edges AF, BG, EK, HC, and DI are perpendicular to the bases. In the right prism, either of the perpendicular edges is equal to the altitude. In the oblique prism the altitude is less than the edge. F B 7. How are prisms distinguished from each other? H A prism whose base is a triangle, is called a triangular prism if the base is a quadrangle, it is called a quadrangular prism: if a pentagon, a pentagonal prism: if a hexagon, it is called a hexagonal prism: &c. |