Albert Irvin Frye.

Civil engineers' pocket book; a reference-book for engineers, contractors, and students, containing rules, data, methods, formulas and tables online

. (page 39 of 182)
Online LibraryAlbert Irvin FryeCivil engineers' pocket book; a reference-book for engineers, contractors, and students, containing rules, data, methods, formulas and tables → online text (page 39 of 182)
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12 8 4 5

men/ *[' ^^ i^ ^ ^ 286 '^ 16384 ^

6 7 etc.

W538 ^ 1048576 •••yjw

Formulas (1) and (2) may be expressed: l^n[s{k)] (3)

in which the continued series A; is a coefficient of s, xnaking sk the diameter
of a circle whose circumference — perimeter of the ellipse.

To facilitate the use of equation (2) : Log x - 0. 4971490; log i - 9.3979400;
kiBA-8.6709413: log ,g« - 8.2907300; log THf4 - 8.0286181; log «HM «
7.S279587; log Ti«ftV« =• 7.6662314. Note that logarithms of e*, e^, e», «">, #"
are 2. 8 6, 6 times log e^.

Problem 1. — ^The major and minor axes of an ellipse are 86 and 24 ft.,
itspectivehr. Find the perimeter?

IRA

So/Klum.— Majoraxis5-36; 0-I8; 6-12; ««-4l? : log««- 9.7447275.

Using 6-place logarithms, we have for the value of k, by terms:
1 2 3 4 6 6 7

Log i-9.39704 A-8070M 8.29073 8.02862 7.82796 7.66623
Log tf^- 9.74473 ^ -9.48946 #< -9.23418 tf*-8.97891 tfio«»8.72364 #" -8.46837
Sum -9.14267 ' 8.16040 7.62491 7.00753 6.65160 6.13360

and numbers corresponding to above logarithms are below:
.Jk-14»000- 0.13889 - 0.01447 - 0.00335 - 0.00102 - 0.00036 - 0.00014
minus the sum of the value of terms abovethe 7th, which, by inspec-
tion, we will assume to equal 0.00007. Hence, * =-0.8417; and the
perimeter /-«*- 3.1416 X 36 X 0.8417-96.194 ft. Ans.



240 n.— MENSURATION.

Method 2. — Let /* perimeter; a « semi-major axis; 6 » semi-minor axis;
^ a+b-



Terms: 12 3 4 5

7



Terms: 12 8

[£* E*
1 + ^ + -^



^2«.4«.e».8«.10» ^2«.4«.e».8«.10«.12« ^J^^



Terms: 12 8 4 6

~" ~ £^ . 25E*



256 " 16384
6 7

49 £W . 441 E"



05636 " 1048576 ^ J ^*^

Formulas (4) and (6) may be expressed: /— K(a+b)K (6)

in which (a+&)/i is the diameter of a circle whose circtLmference» perimeter
of the ellipse.

ToTacilitate the use of equation (5) : Log x— 0.4971499; log 1-9.397400;
log A-8.1938200; log yh- 7.6917600; log 1^^4-7.1835201; log «^^
6.8737162; log ToUiT«- 6.6238387. Note that logarithms of M M £^. £*.
£»o. £" are 2, 4, 6. 8, 10. 12 times log E.

Problem 2. — Solve problem 1 by formula (5) ?

5o/M<iOft.-~a-18: fr-12; a+6-80; a~&-6; £-0.2; log £-9.8010800.
Using 5-place logaritnms, we have for the value of K, by terms:
1 2 3 4 5 7

Log J-9.39794 A-8.19382 7.69176 7.18852 6.87872 0.62384

Log £« -8.60206 £^ °7.20412 £»- 5.80618 £» -4.40824 £>0 -3.01030 £m - 1.61286

Sum-2.00000 6.39794 7.39794 9.69176 11.88402 12.23820
and numbers corresponding to above logarithms are below:

.-.K-l +0.010000+0.000025 + 0.00000026+ + +

-1.0100263; and the perimeter /-;r(a+6)is:- 96.193. Ans.

Comparison of Methods 1 and 2. — A mere glance at the solution of
Problems 1 and 2. illustrating the two preceding methods of calctdating the
perimeter of the ellipse, clearly shows the superiority of Method 2: The 6th.
6th. 7th, etc., terms giving values so small as to be negligible in the present
instance. Moreover, ecjuation (5), with the accompanying logarithmic
values given just below it. will be found quite as rapid to use, in many cases,
as many of our so-called approximate formulas, with, in addition, the
advant£^e of accuracy.



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LENGTHS OF SEMI-ELUPTIC ARCS.



241



17.— Lbnotbb of Sbmi-Blliptic Arcs. A or B

For a— Unity, and for Successive Values of — .

a



Note. — To find A or B: Multiply values of co-
efficient C, in the table, by length of semi-major
axis, or a. Thus, A'-'B'^Ca,

[Calculated from Formula (4-fi).*]




.M


2.00000


.01


2.00061


.tt


2.00193


.03


2.00394


.54


2.00657


.05


2.00971


.08


2.01334


.07


2.01740


.08


2.02188


.09


2.02675


.10


2.03198


.11


2.03757


.12


2.04349


.13


2.04971


.14


2.05624


.15


2.06305


le


2.07014


.17


2.07749


.18


2.08509


.19


2.09293


.20


2.10100


.21


2.10931


.23


2.11782


.23


2.12655


.24


2. 13548


.25


2.14461


.26


2.15392


.27


2.16342


.28


2.17309


.29


2.18294


.30


2.19296


.31


2.20313


.32


2.21347


.33


2.22395



•^ ArcA^ArcB



.00061
.00132
.00201
«00263
.00314
.00363
.00406
.00448
.00487
.00523
.00559
.00592
.00622
.00653
.00681
.00709
.00735
.00760
.00784
.00807
.00831
.00851
.00873
.00893
.00913
.00931
.00950
.00967
.00985
.01002
.01017
.01034
.01048






.33


2.22395


.34


2.23469


.35


2.24537


.36


2.26629


.37


2.26735


.38


2.27854


.39


2.28986


.40


2.30131


.41


2.31288


.42


2.32467


.43


2.33638


.44


2.34831


.45


2.36035


.46


2.37249


.47


2.38475


.48


2.39710


.49


2.40956


.60


2.42211


.51


2.43477


.52


2.44752


.53


2.46036


.54


2.47329


.56


2.48632


.56


2.49943


.67


2.51262


.58


2.52590


.59


2.53926


.60


2.55270


.61


2.56622


.62


2.67982


.63


2.69349


.64


2.60723


.65


2.62105


.66


2.63494



.01064
.01078
.01092
.01106
.01119
.01132
.01145
.01157
.01169
.01181
.01193
.01204
.01314
.01226
.01235
.01246
.01255
.01266
.01275
.01284
.01293
.01303
.01311
.01319
.01328
.01336
.01344
.01352
.01360
.01367
.01374
.01383
.01389



.66


2.63494


.67


2.64890


.68


2.66293


.69


2.67702


.70


2.69118


.71


2.70541


.72


2.71970


.73


2.73406


.74


2.74846


.75


2.76293


.76


2.77747


.77


2.79206


.78


2.80671


.79


2.82141


.80


2.83617


.81


2.85098


.82


2.86584


.83


2.88076


.84


2.89673


.85


2.91075


.86


2.92582


.87


2.94094


.88


2.95611


.89


2.97132


.90


2.98658


.91


3.00189


.92


3.01724


.93


3.03263


.94


3.04807


.95


3.06356


.96


3.07908


.97


3.09466


.98


3.11026


.99


3.12590


1.00


3.14159



Difl.



.01396
.01403
.01409
.01416
.01423
.01429
.01436
.01441
.01447
.01454
.01469
.01465
.01470
.01476
.01481
.01486
.01492
.01497
.01502
.01507
.01612
.01517
.01521
.01526
.01531
.01535
.01539
.01644
.01649
.01562
.01667
.01561
.01564
.01569



* Number of terms used in Formula (4) in calculation of this table:

SOterms for — -0.01; 36. for — - 0.06; 20. for — -0.16; 13. for — - 0.26;
a a a a

7. for— -0.60; 6. for- -0.76; 4. for — -0.90; 8, for— - 0.98; and 2
a <x a a

terms for— — 0.99.



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242



lU—MENSURATION.




Elliptic Segmcat ; and Chord. — - — - -«

Let A >- areas of elliptic segment with chord
C:
B — area of elliptic segment with chord

C;
a " semi-major axis — rad of large circle;
b — semi-minor axis — rad of small circle;
6 — 6 '— rise of segment A ;
a— a*" rise of segment B.

Then, length of chord C. -2a*/ 1- ijj ;

length of chord Ck - 26 Jl- (-) '. - ^-^-''

y ^°^ Fig. 19.

Area segment A : area whole ellipse :: area Mg small circle : area small ciccle.

.'. A — (area seg small circle with same chord CO X -r (I)

o

Area segment B : area whole ellipse :: area seg large circle : area large circle.

.'. B — (area S€g large circle with same chord Ck) X — (3)

(See Tables 7 and 8 of Circular Segments, preceding.)

Problem 1. — ^Pind the area of segment A of the ellipse a— 10, fr — 8,
whose chord is distant ^ — 6 from and parallel with the major axis?

Solution. — Diam of small circle— 16, and middle rise h (— 6— 6') of arc
from chord — 8— 5— 3. Now from Table 8, of Circular Segments, the area

corresponding to ^^^. or .1875.-. 101943 diam> -. 101943 X46S: and

mtdtiplying this value by -r- (see Equation 1) we have.

Area A -.101943 X 4a6 -.101943 X 320 - 82.622. Ans.

It is to be noted that area A — area B when -r- " ""•

a

Problem 2. — ^What is the length of chord C of the ellipse given in
Problem 1?



SoluHon.— From the above formula, C. -2a^/l- /-r-j -20-Jl-^ —



15.612. Arts.



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ELUPTIC SEGMENT, PRISMOIDAL FORMULA.



248



B.— SOLIDS.

. _^, _/« TheoreiB.— If a plane curve / or area a lies whollv on one side
of a strai^t line as axis in its own plane, the surface 5 or volume V gene-
rated by Its whole or partial revolution about that axis is:
5 - / X length of path^ p traversed by cen of grav g of line; or 5 — /^;
K — a X length of path JP traversed by cen of grav G of area; or V-^ar,

"?.: disss w ^ to &} ^«^ ^^ <»^« ^'^pi*^ «^i"*ion.

t -♦2a&,; .-. S - 2x1x9, and xh - S + 2ir/ (1)

P -•2jt3'o; .-. V - 2raXo, and Xo - V + 2«i (2)

Thos, equations (1) and (2) are used for finding the surfaces and vol-
umes of Uie sphere, cone, cylinder, torus (cylindrical ring), paraboloid,
dbpsoid. etc. ; also of their sectors, segments, zones and frustums.

It is to be noted also that these equations enable us to find the centers
(tf gr avity of their lines and areas when their lines, surfaces and volumes
sie known.

PristoMal Forniala^— The volume V of a prismoid is equal to the
length / nmltiplied by the mean area A ; and A is equal to i (sum of end
anas, Ot and 03. +4 times the middle area a«); thus



V-M - j-(ai + 4a.+aa).



(3)

A prismoid is a solid having farallel end faces or areas, joined together
by nguJar surfaces or sides, as tne sides of prisms, cylinders, cones, pyra-
nuds. wedges, or their frustums, or any lateral combination of same. The
Iffismnidal formula will apply also to the sphere, hemisphere and spherical
segment; to warped-surface solids where the warp is continuous between
ends of solid; to railroad cuttings that can be aecomposed into prisms.
wedges, eto. : to two equal cones arranged like an hour glass with bases as
end areas; to the conical wedge botmded on one side by a plane radiating
&om the apex of cone; to the frustums of same; and to many other solids.



18. — Thb Pivb Rboular Polthbdrons.

(AH dihedral or soUd angles are equal, and all faces regtdar polygons. Five

only.)



Namb.


Botmded
by


Tofal
Surfaces
-(ledge)'

tunes


Total
Volume V
-(ledge)'

times


Apothem a,

or radius of

inscribed

sphere.

-ledge

times


Radius r,
or radius of

scribed

sphere,

- ledge

times


leti MifWi r&n. .....


4-^'s


1.7320608


0.1178513


0.2041


0.6124


Ciibe<hexahedron)


CD's


6.0000000


1.0000000


0.6000


0.8660


O^abedron


8^'s


8.4041016


0.4714045


0.4082


0.7071


Dodecahedron


12 0*»


20.6457788


7.6631189


1.1136


1.4013


koeahedron.


20 -^'s


8.6602540


2.1816950


0.7558


0.9611



The volume V of any regular polyhedron is equal to its surface 5 times
ooe-third its apothem a; or, k — iSa;.'. a — 3K-*-5.



*2«-0. 288186.



d by Google



244



n.—MENSURATlON




Fig. 20.

, Prisms and Cylinders. — A frism is a solid with parcUkl ends and paralUl
stdg edges. Hence the ends will be equal and similar polygons (r^iular or
irregular) , and the sides will be parallelograms. A cylinder is a prism with an
infinite number of sides. The ends of the cylinder may be circular, elliptic,
or of any curvature.

• Area. — ^The siuiace of any prism or cylinder, whether right or oblique.
IS equal to the two end areas 4- the perimeter p of any right section s mul-
tiplied by the length / of any lateral element: or S- 2a+p/ (Fig. 20).

Volume. — ^The volume of any prism or cylinder, whether right or oblique,
is equal to the area of any right section s multiplied by the length I of any
lateral element; or V-5 / (Fig. 20).

Also, volume equals area of either end multiplied by the vertical di«t.anoe
between the end faces; or. V— aA (Fig. 20).



>^



Fmstum of Prism or Cylinder. — Prism
or cylinder with end faces not pansdlel
(Fig. 21).

Volume. — ^Let gt^cen of grav of end
area Oi; ^3 of any sectional area at; g^ of
end area a^. Then

V =» axhx ; (hx — vert dist from gz to plane Oj).
V — 03^3 ; (/13 — vert dist from gx to plane 03) .

V — oafcg; <M is vert to plane o^, bet ft and

In general, V^area a of any plane
section multiplied bv the perpendicular
distance h between planes passing through -Jj
centers of gravity of end areas and par-
allel with the said plane section, if a
is a right section Oo. V — a©/. These for-
mulas also enable us to find the relation
between certain elements, as oo/— 01/4 —

Note that Fig. 21 becomes a circular cylindric ungula when the right
section oq is a circle, and hence /"- i (longest side + shortest side).




Fig. 21.



Circular Cylindric Pmstum. — This is a
special case of the preceding in which Oq is a
right circular section whose perimeter is p.

Volume V-ao/-iao(^ + «;

V="a|At; Qix is perp to plane Oj.)
V- 03/13; (/»3 is perp to plane 03.)
AreaA^ax-\-a%-\-pl^ax-\-az-\-p (/i-H/a).




Fig. 22.



LINDERS; CYLINDRICAL WEDGES.



245




lalf-Wedgcs. — The following formulas give the
If -wedges cut from circular cylinders ; /i being the
leasured along the element of the cylinder at Oi.
», (W. (c) and id) — as follows: —

iss than radius r ; lower edge Cf

rea of base at bi) (r—bi) I . (Pig. 23a)

- r^ [C|f- (length of arcaO (r-bi) ]. (Fig. 23a)

radius of cylinder; lower edge — d.

(Fig. 23b)
?-2ffc-(iA. (Fig. 23b)

> r and < diameter d] lower edge >- Cj.

ixcA of base at 62) (&a-o]. (Fig. 23c)

h

> - r- [ C2r+ (2jcf -arc 02) (62-r) 1. (Fig. 23c)

02

diameter of cylinder; lower edge at 02.

(area of circular base). (Pig. 23d)

>->rrfc. (Fig. 23d)

rhether figure is right or oblique. (Fig. 23d)



r right or obliqM figure, h being perp height,
-face, for right figure only. For total surface, add
kd base (circular).



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240



Ih—MENSU RATION,



19.— Propbrtibs of Hollow Ctlindbrs (Pipbs, Tanks or Wbll8).Onb
Foot in Lbngth.

Note that Areas, Volumes, Capacities and Weights are proportional to
the squares of the diameters. 1728 cu. ins. -7.4805 gallons- 1 cu. ft.—
62.6 lbs. (nearly) of water; 231 cu. ins. - 1 gallon; 201. »74 gallons- 1 cu. yd.







Hjrdrau-




Qrcum.










Weight


DIam.


lie Mean


Clroum.




Volume.
Cu. Ins.


Area-


Volume.


Capacity


of






Radius


Ins.


Surface.


Volume.


Cu-Yds.


Gallons.


Water,






d




Ft.




Ft.






Pounds.


ID..


Ft.


"4*












1 ■


.0104


.0026


.392699


.032725


. 147262


.000085


.000003


.00064


.00533


3- 6


.0156


.0039


.589049


.049087


.331340


.000192


.000007


! 00143


.01198




.0208


.0052


.785398


.065450


.589049


.000341


000013


.00265


.02131




.0312


.0078


1.17810


.098175


1.32536


.000767


.000028


.00574


.04794




.0417


.0104


1.57080


.130900


2.35619


.001364


.000051


.01020


.08522




.0521


.0130


1.96350


.163625


3.68155


.002131


.000079


.01594


.13316




.0625


.0156


2.35619


.196350


5.30144


.003068


.000114


.02295


.19175


.0729


.0182


2.74889


.229074


7.21585


.004176


.000155


.03124


.26099


1


.0833


.0208


3.14159


.261799


9.42478


.005454


.000202


.04080


.84088




.1042


.0261


3.92699


.327249


14.7262


.008523


.000316


.06375


.63263


.1250


.0312


4.71239


.392699


21.2058


.012272


.000465


.09180


.76699


li


.1458


.0365


5.49779


.458149


28.8634


.016703


.000619


.12495


1.04396


2


.1667


.0417


6.28319


.523599


37.6991


.021817


.000808


.16320


1.3635


^

2


.1875


.0469


7.06858


.589049


47.7129


.027612


.001023


.20656


1.7267


.2083


.0521


7.85398


.654498


58.9049


.034088


.001263


.25600


2.1305


2f


.2292


.0573


8.63938


.719948


71.2749


.041247


.001528


.30865


2.5779


8


2500


.0625


9.42478


.785398


84.8230


.049087


.001818


.36720


3.0680


3i


.2917


.0729


10.9956


.916298


115.454


.066813


.002475


.49980


4.1769


4


.3333


.0833


12.5664


1.04720


150.796


.087266


.003232


.66280


6.4641


4i


.3750


.0937


14.1372


1.17810


190.852


.110447


.004091


.82620


6.9029


6


.4167


.1042


15.7080


1.30900


235.619


.136354


.005050


1.02000


8.6221


H


.4583


.1146


17.2788


1.43990


285. 100


. 164988


.006111


1.2342


10.31S


6


.5


.1260


18.8496


1.67080


339.292


.196350


.007272


1.4688


12.27S


H


.5417


.1354


20.4204


1.70170


398.197


.230438


.008535


1.7238


14.408


7


.5833


.1458


21.9911


1.83260


461.814


.267254


.009898


1.9992


16. 70S


7i


.6250


.1662


23.5619


1.96350


530.144


.306796


.011363


2.2950


19.175


8


.6667


.1667


25.1327


2.09440


603.186


.349066


.012928


2.6112


31.817


H


.7083


.1771


26.7035


2.22529


680.940


.394063


.014595


2.9478


24.629


9


.7500


.1875


28.2743


2.35619


763.407


.441786


.016362


3.3048


27.61S


H


.7917


.1979


29.8451


2.48709


850.586


.492237


.018231


3.6822


30.765


10


.8333


.2083


31.4159


2.61799


942.478


.545415


.020201


4.0800


84.088


m


.8750


.2187


32.9867


2.74889


1039.08


601320


.022271


4.4982


37.588


11


.9167


.2292


34.5575


2.87979


1140.40


.659953


.024443


4.9368


41.247


l\i


.9583


.2396


36.1283


3.01069


1246.43


.721312


.026715


6.3958


4S.068


12


1.


.25


37.6991


3.14159


1357.17


.785398


.029089


5.8752


49.087


13


1 0833


.2708


40.8407


3.40339


1592.79


.921752


.034139


6.8952


57.609


U


1.1667


.2917


43.9823


3.66519


1847.26


1.06901


.03959


7.9968


66.818


15


1.2500


.3125


47.1239


3.92699


2120.58


1.22718


.04545


9.1800


76.699


16


1.3333


.3333


50.2655


4.18879


2412.74


1.29626


.05171


10.445


87.368


17


1.4167


.3542


53.4071


4.45059


2723.76


1.57625


.05838


11.791


98.518


18


1.5


.375


66.5487


4.71239


3053.63


1.76715


.06545


13.219


110.45


19


1.5833


3958


59.6903 |4. 97419


3402.34


1.96895


.07292


14.729


123.08


20


1.6667


.4167


62.8319 !5. 23599


3769.91


2.18166
2.<Qt981


.08080


16.330


136.35


22


1.8333


.4583


69.1150


5.75959


4561.59


.09777


19.747


164.99


24


2.


.6


75.3982


6.28319


5428.67


3.14159


.11636


23.601


196.35



The Circumference is proportional to the Diameter.



d by Google



ES OF HOLLOW CYLINDERS.



247



s OF Hollow Cylinders. — Concluded.













Weight








ty


of








18.


Water.
Pounds.




1


230.44








7


267.25










306.80










349.07










394.06










441.79










492.24










545.42
601.32










659.95










721.31










785.40










852.21










921.75










994.02










1069.0










1227.2










1484.9










1767.1










2073.9










2405.3










2761.2
3141.6


MV. 1199


v«A4r .o


iM.inia


4.U6U1


«uo.v0


3408.8


28.2743


109931.


63.6173


2.3562


475.89


3976. 1


31.4159


135717.


78.6398


2.9089


687.62


4908.7


84.5576


164217.


95.0332


3.6197


710.90


5939.6


37.6991


195432.


113.097


4.1888


846.03


7068.6


40.8407


229361.


132.732


4.9160


992.91


8295.8


43.9823


266005.


153.938


6.7014


1161.6


9621.1


47. 1239


305363.


176.716


6.5450


1028.2


11045.


50.2656


357435.


201.062


7.4467


1647.3


12566.


53.4071


392222.


226.980


8.4067


1697.9


14186.


56.5487


439722.


254.469


9.4248


1903.6


15904.


59.6903


526938.


283.529


10.501


2276.8


17721.


62.8319


542867.


314.159


11.636


2350.1


19635.


69.1150


656869.


380. 133


14.079


2843.6


23758.


75.8982


781729.


452.389


16.755


3384.1


28274.


78.5898


848230.


490.874


18.181


3672.0


30680.


81.6814


917446.


530.929


19.664


3971.6


33183.


87.9646




615.752
706.858


22.806
26.180


4606.1
6287.7


38484.


94.2478




44179.


100.531




804.248
907.920
1017.88
1134.11
1256.64
1386.44
1520.53


29.787
33.627
37.699
42.004
46.542
51.313
56.316


6116.2
6791.7
7614.3
8483.7
9400.3
10364.
11374.


50265.


106.814




56745.


113.097





63617.


119.381




70882.


125.664




78540.


131.947





865&0.


138.280




95033.


144.513




1661.90
1809.56
1963.50


61.552
67.021
72.722


12432.
13536.
14688.


1038C9.


150.796




113097.


157.080




122719.









d by Google



248



n.^MENSURATION.




Pyramid and Pyramidic Frustum.— A " regu-
lar" pyramid is one in which the base is a regu-
lar polygon; if not, it is " irregular." If the axis,
from the aftx to the cen of grav g of the base,
is perp to the base it is a " nght pyramid ; it
not, It is "oblique." Fig. 24 shows an oblique
pyramid and frustum together forming a right
pyramid.
Volu9H4 Vt of right pyramid — J (area of base X

perp height) — i Oihi.
Volume V* of oblique pyramid — } (area of base

X perp height) — J <h^.

Volume Vt of pyramidic frustum — Vr — V« —

i(aiAi-aafca). „. ^.

Fig. 24.

If Oa is parallel with Ot. appljring the prismoidal formula, volume Vt of
frustum — -~ - 2 (oi + as + Vo|Oa j , whether pyramid is right or oblique.

regular or irregular.

The area of the sides of a right regular pyramid « \ (perimeter of base X
least slant height) * of a right regular frustum with parallel faces — \ (peri-
meter of top + perimeter of base) X least slant height.

For area of an oblique or irregular pyramid or frustum, the sides must be
calculated — as triangles, or as trapezoids or trapesiums, respectively. No
simple general formula will apply.

Center of gravity of pyramid, whether right of oblique, lies in the axis,
and one-fourtn its length from the base.



Coae and Conic Frustum. — A cone may
be considered as a pyramid with an infinite
number of sides. If the axis from the apex
to the cen of grav g of the base, is peri> to
the base it is a ' right" cone; if not, it is
'* oblique." Generally speaking, a right cone
is understood to have a circular base; and
an oblique cone, to have an elliptic base.
Such cones, however, are sometimes termed
right- and oblique circular cones, to distin-
gmsh them from right- and oblique elliptic
cones whose base of right cone is an ellipse.
Note that an oblique circular cone may be
cut from a right elliptic cone; or that an
oblique elliptic cone may be cut from either
a circular or an elliptic cone. Fig. 25 shows
an oblique cone and frustxmi together forming
a right cone.




Fig. 25.



Volume V, of right cone— J (area of base X perp height) - J athi.
Volume Vo of oblique cone— J (area of base X perp height) — J a^.
Volume Vt of conic frustum - V, — V. - i (ojfci — o^fea) .

If at is parallel with at. applying the prismoidal formula, volume Vrof
hx-h2



fnistum — . ' fai+02+ Voiajj .*



The area of the curved stirface (side) of a right cone" J (perim of base X
slant height) ; of a right frustum with parallel faces— Kperim of top + perim
of base) X slant height.



* If the bottom and top faces of the frustum are circles with radii r^
and fa, resp>cctively, then Ui =- irfi' and Oj — ;rr2*. If they are elliptical, use
the formula: area of ellipse — ttoo. in which a — seim^major axis and 6 —
semi-minor axis. »r- 3.1416. Digitized by GoOglc



PYRAMID. CONE. WEDGE. SPHERE.



249



The arfa o£ the curved suiiaqe (side) of an oblique tllipUc cone of height
kf (Pig. 24), and which haf been cut from a right circular com, — ^^^, in

which Jbj — pcrp height. 02 -■area of elliptic base, and r'— Aj ^perpdist

cos ex

irom side of right circular cone to point where axis of same pierces base a^



of obHque cone. Hence, area»-



oa^a _ 02 cos <X
sin;9



Also, area— -7 (volume of



oblique cone)—



ZV.



Center of gravity of cone, whether right or oblique, lies in the axis and
one-fourth its length from the base.



Coaic Wedge and Fmstmn from right cone. — If
the wedge is cut from the cone by a plane pass-
ing through the apex (Pig. 26), with Oa parallel to



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