11. To find the area of a circle. I. Multiply half the circumference by half the diameter. (VI. 12.) II. Multiply the square of the radius by п. (VI. 12, Cor. 1.) 12. To find the area of a sector of a circle. I. Say, As four right angles is to the angle of the sector—or, As the whole circumference is to the arc of the sector—so is the area of the circle to the area of the sector. (V. 25.) II. Multiply the arc of the sector by the radius, and divide by two. (Proved as VI. 12.) EXAMPLES. 1. The adjacent sides AB and AC of a parallelogram are 20 and 16, the distance on AB to the foot of the perpendicular from C is 5; what is the area? Ans. 303.8+. 2. The area of a triangle is 40 and the base is 10; what is the altitude? Ans. 8. 3. The sides of a rectangle are 16 and 25; what is the side of a square containing an equal area? of an equilateral triangle? Ans. 20; 30.39. If a be a side of an equilateral triangle, then area is equal to a2 1/3. 4. What would it cost to fence the above in the three cases at 50 cents per linear foot? Ans. $41; $40; $45.58. 5. A side of a regular hexagon is 10; what is the area? Ans. 259.8. 6. The gable end of a house is 40 feet wide, the height to the eaves is 40 feet, and to the crown 50 feet; how many bricks, the face of each being 8 in. by 2 in., will be required to build it, the wall being 2 bricks thick? Ans. 32,400. 7. The short sides of a right-angled triangle are 25 and 36; what is the other side? what the side of an equivalent square? Ans. 43.8+; 21+. 8. The parallel sides of a trapezoid are 20 and 16, and the perpendicular distance between them is 6; if the other sides be produced to meet, what is the area of the whole triangle formed? Ans. 300. 9. In the last example the trapezoid is bisected by a line parallel to the parallel sides; how far from the longer side must it be drawn? Ans. 2.84+. It will assist in solving this to remember that similar triangles are to each other as the squares of their homologous sides. 10. A man has three lots, each containing 10 acres; one is a square, one a circle, and one a semicircle; which requires the least fencing to enclose it? Ans. The circle. 11. A farmer desiring to know the contents of a quadrilateral field, steps off a diagonal, 200 steps; he then goes to the other corners, and steps perpendiculars to this diagonal, 80 and 30 steps. If each step be 3 feet, how many acres in the field? Ans. 2 A. 1 R. 3+r. 12. In a field AD was found to be 200 chains, Bb 20, Cc 18, Ee 40, Gg 10, Ff 42; also Af was 15, fb 10, bg 15, gc 100, ce 40, and eD 20. What is the area? Ans. 969 A. It will be noted that the area is +AGg-FfgG. D E B 9 b A 13. One side of a regular polygon is 25, and an angle is 108°; what is the area of the polygon? Ans. 1075.277. 14. If the diameter of the earth be 7912 miles, what is the circumference? Ans. 24856.28. 15. If 15° of the equator pass under the sun in one hour, how many miles will pass in 2 hours and 22 minutes? Ans. 2451.097 miles. 16. What is the length of one degree of the equator? Ans. 69.045 miles. 17. A circular plot 80 feet in diameter has a walk 8 feet wide running around it, inside the boundary; what amount of land is taken up by it? Ans. 1809.56. 18. A chord of a circle is 80, and its distance from the centre is 30; what is the area of the circle? Ans. 7854. 19. What is the side of a square equivalent to a circle 60 feet in diameter? Ans. 53.17. 20. A rectangle is 10 by 7; find the radius of a circle of the same perimeter. Ans. 5.41. 21. The hypotenuse of a right-angled triangle inscribed in a circle is 40; what is the area of the circle? Ans. 400π. 22. From a point without a circle a line of 8 feet is drawn touching the circle, and a line of 4 feet meeting the circle at its nearest point; what is the area of the circle? Аns. 36π. 23. What is the area of a sector of 25° in a circle whose radius is 44? Ans. 422.367. 24. What is the area of one of the segments cut off by a side of an inscribed square in a circle whose radius is 10? Ans. 28.54. 25. Two chords on the same side of the centre of a circle subtend angles of 60° and 40°; the radius is 14 feet; what is the area of the zone? Ans. 11.361 sq. f .ft. 26. A ladder 100 feet long is placed against a perpendicular wall; how far will the top move down by moving the bottom of it 20 feet from the wall? Ans. 2.03 ft. 27. A side of a square is 20 feet; find the difference between the areas of the inscribed and circumscribed circles. Ans. Area of smaller circle. 28. Two circles, each having a radius of 10 feet, cut each other; and the angle at the centre of either, between lines drawn to the points of section, is 40°; what are the areas of the three figures formed? Ans. 308.626; 5.534; 308.626. 29. A field is 200 metres long and 50 wide; what is its area? Ans. 1 hectare. 30. Give the three dimensions of this book in metric units. Calculate its surface. 31. A pole 2 metres high casts a shadow 2.6 metres long; what is the height of a tree whose shadow at the same instant is 78 metres ? Ans. 60 metres. 32. A rectangle 5 centimetres long and 2 wide has a semicircle described on its two ends; what is the area of the whole figure? Ans. 13.1416 square centimetres. 33. A horse can pasture over a circle 20 metres in radius; how much Ans. 12.5664 ares. land in the circle? PART II. SOLID GEOMETRY. BOOK VII. PLANES. DEFINITIONS. 1. A plane or plane surface is one in which, if any two points be taken, the straight line joining them lies wholly in that surface. It has no thickness, but is indefinite in extent in the other two directions. 2. A straight line is perpendicular to a plane when it makes a right angle with every straight line which it meets in that plane. 3. A straight line is parallel to a plane when, being produced ever so far, it does not meet it. 4. The projection of a point on a plane is the foot of the perpendicular drawn from the point to the plane. 5. The projection of a line on a plane is the line formed by the projection of all its points. |