MENSURATION. 277. A triangle is a figure bounded by three straight lines. Thus, BAC is a triangle. The three lines BA, AC, BC, С are called sides : and the three corners, B, A, and C, are called angles. The side AB is called the base. When a line like CD is dráwn A making the angle CDA equal to the angle CDB, then CD is said to be perpendicular to AB, and CD is called the altitude of the triangle. Each triangle CAD or CDB is called a right-angled triangle. The side BC, or the side AC, opposite the right angle, is called the hypothenuse .. The area or content of a triangle is equal to half the product of its base by its altitude. EXAMPLES. OPERATION. 1. The base, AB, of a triangle is 40 yards, and the perpendicular, CD, 20 yards : what is the area ? We first multiply the 40 base by the altitude, and 20 the product is square yards, which we divide by 2 for 2)800 the area. Ans. 400 square yards. 2. In a triangular field the base is 40 chains, and the perpendicular 15 chains : how much does it contain ? (Art. 87). Ans. 30 acres. 277. What is a triangle ? What is the base? What the altitude ? What is a right-angled triangle? Which side is the hypothenuse ? What is the area of a triangle equal to ? 3. There is a triangular field, of which the base is 35 rods and the perpendicular 26 rods: what is its content ? Ans. 2A. 3R. 15P. 278. A square is a figure having four equal sides, and all its angles right angles. 279. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. D 280. A parallelogram is a foursided figure which has its opposite sides equal and parallel, but its angles not right angles. The line DE, perpendicular to the base, is called the altitude. , A E B 281. To find the area of a square, rectangle, or paral. lelogram, Multiply the base by the perpendicular height, and the product will be the area. EXAMPLES. a 1. What is the area of a square field of which the sides are each 33.08 chains ? Ans. 109A. 1R. 28P.+. 2. What is the area of a square piece of land of which the sides are 27 chains ? Ans. 3. What is the area of a square piece of land of which the sides are 25 rods each ? Ans. 3A. 3R. 25P. 278. What is a square ? 231. How do you find the area of a square, rectangle, or parallelogram? a 4. What is the content of a rectangular field, the length of which is 40 rods and the breadth 20 rods? Ans. 5 acres. 5. What is the content of a field 40 rods square ? Ans. 10 acres. 6. What is the content of a rectangular field 15 chains long and 5 chains broad ? Ans. 7. What is the content of a field 27 chains long and 9 rods broad ? Ans. 6A. OR. 12P. 8. The base of a parallelogram is 271 yards, and the perpendicular height 360 feet : what is the area ? Ans. 32520 square yards. a 282. A trapezoid is a four-sided fig. ure ABCD, having two of its sides, AB, DC, parallel. The perpendicular CE is called the altitude. Е В 283. To find the area of a trapezoid, Multiply the sum of the two parallel sides by the alti. tude, and divide the product by 2, the quotient will be the area. EXAMPLES. 1. Required the area of the trapezoid ABCD, having given AB=321.51ft., DC=214.24ft., and CE=171.16ft. OPERATION. We first find the 321.51 +214.24=535.75= sum of the sides, and sum of parallel sides. then multiply it by the Then, perpendicular height, 535.75 X 171.16=91698.97 after which, we divide 91698.97 the product by 2, for and, =45849.485 the area. 2 —the area. 282. What is a trapezoid ? 2. What is the area of a trapezoid, the parallel sides of which are 12.41 and 8.22 chains, and the perpendicu. lar distance between them 5.15 chains ? Ans. 5A. IR. 9.956 P. 3. Required the area of a trapezoid whose parallel sides are 25 feet 6 inches, and 18 feet 9 inches, and the perpendicular distance between them 10 feet and 5 inches. Ans. 230 Sq. ft. 5' 7". 4. Required the area of a trapezoid whose parallel sides are 20.5 and 12.25, and the perpendicular distance between them 10.75 yards. Ans. 176.03125 Sq. yd. 5. What is the area of a trapezoid whose parallel sides are 7.50 chains, and 12.25 chains, and the perpendicular height 15.40 chains ? Ans. 15A. OR. 33.2P. 6. What is the content when the parallel sides are 20 and 32 chains, and the perpendicular distance between them 26 chains ? Ans. 67 A. 2R. 16P 284. A circle is a portion of a plane bounded by a curved line, every part of which is equally distant from a certain point within, called the centre. C The curved line A EBD is called the circumference; the point C the E centre ; the line AB passing through the centre, a diameter ; and CB the radius. The circumference AEBD is 3.1416 times greater than the diameter AB. Hence, if the diameter is 1, the circumference will be 3.1416. Hence, also, if the diameter is known, the circumference is found by mul. tiplying 3.1416 by the diameter. 284. What is a circle? What is the centre? What is the diameter? What the radius? How many times greater is the circumference than the diameter? How do you find the circumference when the diameter is known ? EXAMPLES. 1. The diameter of a circle is 4, what is the circumference ? OPERATION. The circumference is found 3.1416 by simply multiplying 3.1416 4 by the diameter. Ans. 12.5664 2. The diameter of a circle is 93, what is the circumference ? Ans. 3. The diameter of a circle is 20, what is the cir. cumference ? Ans. 62.832. 285. Since the circumference of a circle is 3.1416 times greater than the diameter, it follows that if the circumference is known we may find the diameter by dividing it by 3.1416. EXAMPLES. a 1. What is the diameter of a circle whose circumfer. ence is 78.54. We divide the circumference by 3.1416, the quotient 25 is the diameter. OPERATION. 3.1416)78.5400(25 62832 2. What is the diameter of a circle whose circumfel a ence is 11652.1944 ? Ans. 3709. 3. What is the diameter of a circle whose circumfer. ence is 6850 ? Ans. 2180.41+. 286. To find the area or content of a circle, Multiply the square of the diameter by the decimal .7854. 285. How do you find the diameter when the circumference is known? 286. How do you find the area of a circle ? |