LESSONS IN MENSURATION. 



77 



KXERCIHB 2. 



1. Tho base of a triangle ia 6, and its two aidee 5 and 7 ; 

 what in it* altitude P 



i ho bane being 8, and the two side* 10 and 6; required tho 

 altiti. 



i ho three aidea of A triangle are 21, 20, and 13 ; what i 

 its perpondioulur P 



4. The base of a triangle ia 5-9(5, and ita altitndo 3-81 ; what 

 is its urr.ir 



5. Tho base of a triangle is 7'37 chains (1 chain = G6 foot), 

 uiul (lit- iilntu.lo 4'98 chains ; what is its area P 



6. Tho hypothonuso of a right-angled triangle is 205, and the 

 base 200 ; required tho area '( 



7. Tho side of an oquilatoral triangle in 3'-t ; what is its area P 



8. Suppose tho base of an isosceles triangle, whoso area ia one 



foot ; what is its altitude P 



Wo subjoin another rule for tho calculation of the area of a 

 trianglo without finding its perpendicular, tho three sides being 

 given: From half tho sum of the three sides subtract each 

 side separately. Then multiply the half sum by tho three 

 remainders successively, and tho square root of tho product will 

 be tho area. 



EXAMPLE 1. The three sides of a triangle are 13, 20, and 

 21 ; find its area by tho above rule. 



13 27 27 27 27 



20 13 20 21 6 



21 



14 7 6 162 



2)54 7 



27 



1134 

 14 



4536 

 1134 



V15876 = 126 Ant. 

 EXERCISE 3. 



1. The three sides being 13, 14, 15, what is tho area ? 



2. The side o ~ % hexagon is 10 ; what is its area computed 

 by both the foregoing rules ? 



3. The side of the base of a square pyramid measures 12 ft., 

 and the perpendicular height 10 ft. ; what is tho length of the 

 slanting edge, and what tho superficial area of the pyramid base 

 included P 



We shall now take our readers a step higher in our subject, 

 ,. but must of necessity introduce some matters 

 which belong more particularly to Trigonome- 

 try, and to our papers on that subject we must 

 direct his attention, for an explanation oi 

 those points ho is unable to understand with- 

 out it. 



PROBLEM I. Let ABC (Fig. 13) be a tri- 

 angle, right-angled at B. Given tho hypo- 



E - 

 Fig. 13. 



thorny* A c, and the angle CAB; required the length of the 

 perpenuiCular B c. Rule : Multiply A c by the natural sine * of 

 the angle CAB; the result will be tho length of c B. 



Let H = the hypothenuse, P = the perpendicular, and s 

 = tho natural sine ; then 



COR. Since P = H X s, ir = 



and s = . 

 H 



EXERCISE 4. 



1. Tho hypothenuse of a right-angled triangle is 10'47, and 

 the angle at tho base 58" 20' ; what is tho height of the perpen- 

 dicular ? 



2. At what anglo do we ascend a regular acclivity 6 miles 

 long, attaining an altitude at tho summit of 4268 feet ? 



3. The hypottsnuso of a right-angled triangle is 89 yards 2 

 feet, and the angle at tho base is 55 ; what is the length of 

 the base P 



* In Fig. 13, on A c measure A n = 1 on a scale of equal parts, 

 and let full from n tho perpendicular i> E on tho base A it, then n E 

 natural sine of tbo angle CAB. To save time, the student should 

 be furnished with a scale of sines, tangents, etc., for reference, for all 

 the angles of the quadrant (90) to within one minute. He will 

 frequently require to refer to the table in calculations in Mensuration. 



!'. 



PROBLEM II. To find the radius of a circle inscribed in 

 given triangle. Uule : Divide twice the area of the triangle by 

 the ram of ita three aidea. 



Let A B c (Fig. 14) be a triangle whoae three sides are given, 

 it is required to find the length of i> . find the ana of to* 

 triangle from previous rnlea, double it, divide 

 the remit bvAB-pBC + CA; the quotient 

 will give D K. 



EXERCISE 8. 



1. The aide of an equilateral triangle ia I0 k 

 what is tho radius of the inscribed circle P 



2. The two legs of a right-angled triangl 

 are 3 and 4 ; what ia tho radioa of the inscribed 

 circle P 



3. The three aides of a tritngle, are 39, 60. 

 and 63 ; what ia the diameter of the inscribed circle ? 



PROBLEM III. The aide of a regular polygon (aee Definitions 

 in "Geometry ") being given, to find tho radii of the circumscribed 

 and inscribed circles. Rule : Divide 860 (the number of de- 

 grees in the whole circumference) by 

 the number of sides in the polygon ; 

 the quotient will be the angle at 

 tbo centre. 



Let A B D K (Fig. 15) be a re- 

 gnlar hexagon, of which the side 

 A B is known ; then by above rule 

 the angle A o B is found. Halve 

 this for A o c, and join o c. Then 

 o c is perpendicular to A B, and 

 bisects it, and the anglo AGO 

 is a right angle. Hence in the 

 right angled triangle A c o, we 

 have given the angle A o c == 



A O B 



and the anglo o A c = 90 - A o c, 



Pig. 15. 



also the perpen- 



A o = 



,. , A B 



uicular A c = - , 



a 



O c = A o X sine o A c. 



EXAMPLE 1. The side of a regular pentagon (five-sided) is 

 1 5 yards ; what are the radii of the circumscribed and inscribed 

 circles ? 



360 -4- 5 = 72 = angle A o B (Fig. 15). 

 72 -7- 2 = 36 = angle A o c ; sine 36 = '5878 ; 

 90 - 36 = 54 = angle c A o ; sine 54 = '8090 ; 



15-25 (A B) 7-625 (AC) 

 then A o = jj or TggTjj- ' J 3 g . = 12*972, about, 



and c o =~ 12'972 (_ ;>) X '8090 (sine 54) = 10-495, about 

 EXERCISE 6. 



1. Tho sido of on octagon is 138 yards ; what are the radii of 

 the circumscribed and inscribed circles respectively ? 



2. The radius of a circle is 1'84; what will be the length of 

 tho side of a heptagon inscribed in it, and of an equilateral 

 triangle describ".^ <*bout it r 



3. If 1 inch is tho distance between the opposite sides of a 

 decagon, what is the distance between its opposite angles ? 



Wo now come to the consideration of tho relations which exist 

 between the various lines connected with circles ; and first of 

 the proportion between the circumference and the diameter. 

 This may bo regarded approximately as 22 to 7, but more 

 correctly as 3'1416 to 1*. Hence if D = diameter, c = cir- 

 cumference and = S'1416, 



t> = -- and c = D T, 



EXERCISE 7. 



1. Tho diameter of a circle is S ; what is its circumference? 



2. What is the circumference of the earth, supposing its dia 

 meter is 7958 miles P 



8. Tho circumference of the earth at the equator being 2489* 

 miles, what is its equatorial diameter P 



4. What is the circumference of a circle whose radius is 2| 

 feet? 



* The exact proportion between the circumference of a circle to 

 its diameter, the diameter being 1, has never been found. It may be 

 continued to more than 100 places of decimal*. 



