2. What is the area of a zone, one side of which is 23.25, and the other side 20.8, in a circle whose diameter is 24? Ans. 208. 38. If the diameter of the circle is not given, it may be found from the sides and the breadth of the zone. Let the center of the circle be at O. (Fig. 12.) Draw ON perpendicular to AH, NM perpendicular to LR, and HP perpendicular to AL. Then AN AH, (Euc. 3. 3.) MN=(LA+RH) The triangles APH and OMN are similar, because the sides of one are perpendicular to those of the other, each to each. Therefore PH: PA::MN: MO MO being found, we have ML-MO=OL. 2 1.) Ex. If the breadth of the zone ACDH (Fig. 12.) be 6.4, and the sides 6.8 and 6; what is the radius of the circle? PA 3.4-3=0.4. And MN=(3.4+3)=3.2. Then 6.4 0.4::3.2: 0.2=MO. And 3.2-0.2=3=OL And the radius CO=√32+(3.4)2=4.534. PROBLEM VIII. To find the area of a LUNE or crescent. 39. FIND THE DIFFERENCE OF THE TWO SEGMENTS WHICH ARE BETWEEN THE ARCS OF THE CRESCENT AND ITS CHORD. If the segment ABC (Fig. 14.) be taken from the segment ABD; there will remain the lune or crescent ACBD. Ex. If the chord AB be 88, the height CH 20, and the height DH 40; what is the area of the crescent ACBD? To find the area of a RING, included between the peripheries of two concentric circles. 40. FIND THE DIFFERENCE OF THE AREAS OF THE TWO CIRCLES. Or, Multiply the product of the sum and difference of the two diameters by .7854. The area of the ring (Fig. 13.) is evidently equal to the difference between the areas of the two circles AB and A'B'. But the area of each circle is equal to the square of its diameter multiplied into .7854. (Art. 30.) And the difference of these squares is equal to the product of the sum and difference of the diameters. (Alg. 235.) Therefore the area of the ring is equal to the product of the sum and difference of the two diameters multiplied by .7854. Ex. 1. If AB (Fig. 13.) be 221, and A'B' 106, what is the area of the ring? 2 2 Ans. (221 ×.7854)-(106 ×.7854)=29535. 2. If the diameters of Saturn's larger ring be 205,000 and 190,000 miles, how many square miles are there on one side of the ring? Ans. 395000 X 15000 X.7854 4,653,495,000. PROMISCUOUS EXAMPLES OF AREAS. Ex. 1. What is the expense of paving a street 20 rods long, and 2 rods wide, at 5 cents for a square foot? 4 Ans. 544 dollars. 2. If an equilateral triangle contains as many square feet as there are inches in one of its sides; what is the area of the triangle? Let x=the number of square feet in the area. x Then = the number of linear feet in one of the sides. 12 3. What is the side of a square whose area is equal to that of a circle 452 feet in diameter ? Ans. √(452) X.7854=400.574. (Art. 30 and 7.) 4. What is the diameter of a circle which is equal to a square whose side is 36 feet? Ans. √(36)2÷0.7854=40.6217. (Art. 4. and 32.) 5. What is the area of a square inscribed in a circle whose diameter is 132 feet? Ans. 8712 square feet. (Art. 33.) 6. How much carpeting, a yard wide, will be necessary to cover the floor of a room which is a regular octagon, the sides being 8 feet each? Ans. 34 yards. 7. If the diagonal of a square be 16 feet, what is the area? Ans. 128 feet. (Art. 14.) 8. If a carriage wheel four feet in diameter revolve 300 times, in going round a circular green; what is the area of the green? Ans. 4154 sq. rods, or 25 acres, 3 qrs. and 34 rods. 9. What will be the expense of papering the sides of a room, at 10 cents a square yard; if the room be 21 feet long, 18 feet broad, and 12 feet high; and if there be deducted 3 windows, each 5 feet by 3, two doors 8 feet by 4, and one fire-place 6 feet by 41? Ans. 8 dollars 80 cents. 10. If a circular pond of water 10 rods in diameter be surrounded by a gravelled walk 8 feet wide; what is the area of the walk? Ans. 16 sq. rods. (Art. 40.) 11. If CD (Fig. 17.) the base of the isosceles triangle VCD, be 60 feet, and the area 1200 feet; and if there be cut off, by the line LG parallel to CD, the triangle VLG, whose area is 432 feet; what are the sides of the latter triangle? Ans. 30, 30, and 36 feet. 12. What is the area of an equilateral triangle inscribed in a circle whose diameter is 52 feet? Ans. 878.15 sq. feet. 13. If a circular piece of land is enclosed by a fence, in which 10 rails make a rod in length; and if the field contains as many square rods, as there are rails in the fence; what is the value of the land at 120 dollars an acre? Ans. 942.48 dollars. 14. If the area of the equilateral triangle ABD (Fig. 9.) be 219.5375 feet; what is the area of the circle OBDA, in which the triangle is inscribed? The sides of the triangle are each 22.5167. (Art. 11.) 530.93. 15. If 6 concentric circles are so drawn, that the space between the least or 1st, and the 2d is between the 2d and 3d between the 3d and 4th between the 4th and 5th 21.2058, 35.343, 49.4802, 63.6174, 77.7546; between the 5th and 6th what are the severa! diameters, supposing the longest to be equal to 6 times the shortest ? Ans. 3, 6, 9, 12, 15, and 18. 16. If the area between two concentric circles be 1202.64 square inches, and the diameter of the lesser circle be 19 inches, what is the diameter of the other? 17. What is the area of a circular segment, whose height is 9, and base 24? 28 SECTION III. SOLIDS BOUNDED BY PLANE SURFACES, ART. 41. DEFINITION I. A prism is a solid bounded by plane figures or faces, two of which are parallel, similar, and equal; and the others are parallelograms. II. The parallel planes are sometimes called the bases or ends; and the other figures, the sides of the prism. The lat ter taken together constitute the lateral surface. III. A prism is right or oblique, according as the sides are perpendicular or oblique to the bases. IV. The height of a prism is the perpendicular distance between the planes of the bases. In a right prism, therefore, the height is equal to the length of one of the sides. V. A Parallelopiped is a prism whose bases are parallelograms. VI. A Cube is a solid bounded by six equal squares. It is a right prism whose sides and bases are all equal. VII. A Pyramid is a solid bounded by a plane figure called the base, and several triangular planes, proceeding from the sides of the base, and all terminating in a single point. These triangles taken together constitute the lateral surface. VIII. A pyramid is regular, if its base is a regular polygon, and if a line from the center of the base to the vertex of the pyramid is perpendicular to the base. This line is called the axis of the pyramid. IX. The height of a pyramid is the perpendicular distance from the summit to the plane of the base. In a regular pyramid, it is the length of the axis. X. The slant-height of a regular pyramid, is the distance. from the summit to the middle of one of the sides of the base. XI. A frustum or trunk of a pyramid is a portion of the solid next the base, cut off by a plane parallel to the base. The height of the frustum is the perpendicular distance of the two parallel planes. The slant-height of a frustum of a |