LESSONS IN MKNSTK \TION. 



2. A square table is 3 foot across either aide ; bow many 

 squares of 1 inch could bo marked out upon 



\ deal board U 11 foot long and 11 inches wi< 

 endn being square. I want to out it up into piece* 1 foot long 

 ;u,il I in. h \M .! ; how many can I out? 



i The area of a square field is 1 acre ; how long is each side 

 In links and yards P 



5. A rectangular space, intended for planting, is 300 yards 

 long and 220 yards broad. If I cut a path across it the longest 



wide, how much space will remain available for 

 planting P 



6. A street, 30 feet wido and 1 milo long, has to be paved at 

 a cost of 4s. per square yard ; what will the total cost be P 



ho adjacent sides of an acute-angled parallelogram are 20 

 feet ; at what angle must they incline so that the area of 

 the parallelogram shall be 259-8 feet ? 



A hat would be tho area of the above figure if the angle 

 wore 30 instead of 60 P 



PROBLEM IX. The diagonal of a square being given, to find 

 its area. Rule : Square the diagonal and halve the product. 

 Hence the side of a square is to the diagonal as the square root 

 of half its square. 



EXAMPLE 1. Tho diagonal of a square is 10; what is its 

 utv:i f 



102 = 100 ; *J? = 50, area. 



2 



EXAMPLE 2. What is the length of the side of the above 

 square P 



Area = 50 ; \/50 = about 7'07. 



Henco, approximately, the side of a square is to its diagonal 

 as 7 to 10. 



EXERCISE 13. 



1. The area of a square is 100 square feet ; what is tho length 

 of the diagonal ? 



J. Tho diagonal of a square is 4 chains ; required the area of 

 the square. 



3. The area of a square is 1 acre, 1 rood ; what is the length 

 of the diagonal ? 



PROBLEM X. To find tho area of a triangle, the base and 

 A E altitude being given. Let ABC (Fig. 



7*19) bo a triangle, and A D its altitude. 

 f / Rule : Multiply tho base by half the 

 /'' altitude ; tho product is the area. 



The truth of this is evident from 

 Euclid I. 41, in which it is proved 



'Fig. 19. c 



that if a triangle and a parallelogram be upon the same 

 base, and between the same parallels, the triangle is one-half 

 the parallelogram ; for the area of tho parallelogram A B c E is 



B c X AD; and hence that of tho triangle ABcisscX ^- . 



2 



EXAMPLE. Tho base of a triangle is 20, and the altitude 

 is 10 ; what is tho area of the triangle ? 



20 X ^- = 100, area of triangle. 

 2 



EXERCISE 14. 



1. The base of a triangle is 43, and tho altitude 21 ; required 

 tho area. 



2. The base of a triangle is 150 yards, and the altitude 120 

 yards ; required the area in acres, roods, etc. 



3. The hypothenuso of a right-ang^d 

 triangle is 68, and the base 24 ; what is tho 

 area ? 



4. The side of an equilateral triangle is 6 ; 

 what is its area ? 



5. The three sides of a triangle are re- 

 spectively 20, 21, and 29 polos ; required its 

 area in acres, roods, and poles. 



PROBLEM XL To find tho area of a 

 trapezium.* Rule : Divide the figure into 

 two triangles, by drawing a diagonal ; then compute the areas 

 of the triangles separately, by previous rules, and add these 

 areas together. 



Tig. 20. 



A trapezium is a quadrilateral figure (four-sided) in which no two 

 of its sides are parallel. The problem may be extended to finding the 

 area of any quadrilateral figure. 



The oorrectnoM of the working may be prorM by drawing 

 the oppotite diagonal, and repeating the computation. The two 

 results will agree if the calculation U correct. 



EXAMPLE 1. The length* of the four side* of a trapezium 

 A B c D (Fig. 20) are a* follow : A B *= 20 ; B c = 12 ; CDS?; 

 and D A = 18, and the diagonal B D u 18. What 'u it* area ? 



In tho triangle A B D, the two sides A D and o B being equal, 

 its area in doable that of A B D. 



15, nearly, 

 or 10 x 7'5 = 75 ; ar.4 



AEX 



i. i> 



Therefore area of A t D > 



= 2 All* or 150. 

 Again, in tho triangle BCD, the area it found as follows, too 

 three sides being given : 



BC + CD + DB _ 12 -f 7 + 18 _ 37 m 18 . 5 



2 2 ~ = 2 



18-5-12 = 6-5(a); 18-5-7 = 11-5(1); 18'5- 18 - -5{c). 

 18-5 



120-25 



11-5(6) 

 1382-875 



-5(e) 



<:: 1 r,7.', Area of B c D = V691-4375 = 26'3, nearly. 

 Then area of trapezium = areas of ABD + BCD or 150 

 + 26-3 = 176-3. Ans. 



In actual measurements, the diagonal A c may be ascertained, 

 and the areas of the two triangles ABC and A C D found. Their 

 sum will be. found to be as above, provided the measurement. 

 and calculations are correctly performed. 



Note. In tho application of this, and any other rule for the 

 measurement of surfaces as applied to land surveying, too 

 many checks on tho correctness of the results cannot be taken. 



EXERCISE 15. 



1. The four sides of a trapezium are respectively 20, 16, 12, 

 and 14 ; the diagonal across between the two most obtuse angles 

 (draw the figure) is 14. Required the area of the trapezium. 



2. The four sides of a trapezium are C28, 464, 457, and 733, 

 and tho diagonal from the angle between the two shortest sides 

 the opposite angle is 835. Required its area. 



To this as well as to the other exercises that have been given 

 as necessary appendages to the different problems, the learner 

 can easily add examples for practice by substituting other num- 

 bers in the various examples in each exercise ; or by drawing 

 triangles, parallelograms, trapeziums, etc., according to scale, 

 and working out their contents for their dimensions. 



KEY TO EXERCISES IN LESSONS IN MENSUEATION.-IL 

 EXERCISE 1. 



1. 5 ft. 10-7 in., approximately. 



2. 855 feet. 



3. 1 foot 2 itches. 



4. 5 ft. 11 .J in., nearly. 



5. 44 feet. 



6. -866025, etc. 



1. 4-899 nearly. 



2. 6. 



3. 12. 



6. 4500. 



7. About 5-005J. 



8. 240 feet 



3. About 5 4 -4A 



EXERCISE 2. 



4. 11-3538. 



5. Nearly 1 acre, 3 



roods, ISA poles. 



EXERCISE 3. 



1. 84. | 2. 259-8. 



3. Length of edge, 13'11 feet ; area of pyramid, 12376 feet. 



EXERCISE 4. 



1. About 8-91. | 2. 7 9*. 



EXERCISE 5. 

 1. 2-8868. | 2. 1. 



EXERCISE 6. 



1 Nearly 180-5, and about 166 5. | 2. Nearly 1'597, and nearly 6 ! 374. 

 3. About 1-051. 



EXERCISE 7. 



1 25-1327. I Ab ot ls>06t - 



2 25000-8528 miles. 5. Nearly 1 foot *53 inches 

 S.' Nearly 7925. I Nearly 1123 milea. 



