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)
Font size 12 8 4 5

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

6 7 etc.

W538 ^ 1048576 â€˘â€˘â€˘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 â€” perimeter of the ellipse.

To facilitate the use of equation (2) : Log x - 0. 4971490; log i - 9.3979400;
kiBA-8.6709413: log ,gÂ« - 8.2907300; log THf4 - 8.0286181; log Â«HM Â«
7.S279587; log TiÂ«ftVÂ« =â€˘ 7.6662314. Note that logarithms of e*, e^, eÂ», Â«">, #"
are 2. 8 6, 6 times log e^.

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

IRA

So/Klum.â€” Majoraxis5-36; 0-I8; 6-12; Â«Â«-4l? : logÂ«Â«- 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 tfioÂ«Â»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-14Â»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 /-Â«*- 3.1416 X 36 X 0.8417-96.194 ft. Ans.

240 n.â€” MENSURATION.

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

Terms: 12 3 4 5

7

Terms: 12 8

[ÂŁ* E*
1 + ^ + -^

^2Â«.4Â«.eÂ».8Â«.10Â» ^2Â«.4Â«.eÂ».8Â«.10Â«.12Â« ^J^^

Terms: 12 8 4 6

~" ~ ÂŁ^ . 25E*

256 " 16384
6 7

49 ÂŁW . 441 E"

05636 " 1048576 ^ J ^*^

Formulas (4) and (6) may be expressed: /â€” K(a+b)K (6)

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

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

Problem 2. â€” Solve problem 1 by formula (5) ?

5o/M<iOft.-~a-18: fr-12; a+6-80; a~&-6; ÂŁ-0.2; log ÂŁ-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 ÂŁÂ« -8.60206 ÂŁ^ Â°7.20412 ÂŁÂ»- 5.80618 ÂŁÂ» -4.40824 ÂŁ>0 -3.01030 ÂŁm - 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. â€” 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

LENGTHS OF SEMI-ELUPTIC ARCS.

241

17.â€” Lbnotbb of Sbmi-Blliptic Arcs. A or B

For aâ€” Unity, and for Successive Values of â€” .

a

Note. â€” 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

â€˘^ ArcA^ArcB

.00061
.00132
.00201
Â«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 â€” -0.01; 36. for â€” - 0.06; 20. for â€” -0.16; 13. for â€” - 0.26;
a a a a

7. forâ€” -0.60; 6. for- -0.76; 4. for â€” -0.90; 8, forâ€” - 0.98; and 2
a <x a a

terms forâ€” â€” 0.99.

242

lUâ€”MENSURATION.

Elliptic Segmcat ; and Chord. â€” - â€” - -Â«

Let A >- areas of elliptic segment with chord
C:
B â€” area of elliptic segment with chord

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

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

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

y ^Â°^ Fig. 19.

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

.'. A â€” (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 â€” (area Sâ‚¬g large circle with same chord Ck) X â€” (3)

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

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

Solution. â€” Diam of small circleâ€” 16, and middle rise h (â€” 6â€” 6') of arc
from chord â€” 8â€” 5â€” 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 â€” area B when -r- " ""â€˘

a

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

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

15.612. Arts.

ELUPTIC SEGMENT, PRISMOIDAL FORMULA.

248

B.â€” SOLIDS.

. _^, _/Â« TheoreiB.â€” 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 â€” /^;
K â€” a X length of path JP traversed by cen of grav G of area; or V-^ar,

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

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

P -â€˘2jt3'o; .-. V - 2raXo, and Xo - V + 2Â«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^â€” 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 aÂ«); 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. â€” 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,

inscribed

sphere.

-ledge

times

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*Â»

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 â€” iSa;.'. a â€” 3K-*-5.

*2Â«-0. 288186.

244

n.â€”MENSURATlON

Fig. 20.

, Prisms and Cylinders. â€” 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.

â€˘ Area. â€” ^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. â€” ^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 diÂ«t.anoe
between the end faces; or. Vâ€” aA (Fig. 20).

>^

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

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

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

V â€” 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 â€” aÂ©/. These for-
mulas also enable us to find the relation
between certain elements, as oo/â€” 01/4 â€”

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. â€” This is a
special case of the preceding in which Oq is a
right circular section whose perimeter is p.

Volume V-ao/-iao(^ + Â«;

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

Fig. 22.

LINDERS; CYLINDRICAL WEDGES.

245

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

iss than radius r ; lower edge Cf

rea of base at bi) (râ€”bi) I . (Pig. 23a)

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

radius of cylinder; lower edge â€” 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).

240

Ihâ€”MENSU RATION,

19.â€” 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.â€”
62.6 lbs. (nearly) of water; 231 cu. ins. - 1 gallon; 201. Â»74 gallons- 1 cu. yd.

Hjrdrau-

Qrcum.

Weight

DIam.

lie Mean

Clroum.

Volume.
Cu. Ins.

Area-

Volume.

Capacity

of

Ins.

Surface.

Volume.

Cu-Yds.

Gallons.

Water,

d

Ft.

Ft.

Pounds.

ID..

Ft.

"4*

1 â–

.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.

ES OF HOLLOW CYLINDERS.

247

s OF Hollow Cylinders. â€” 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

vÂ«A4r .o

iM.inia

4.U6U1

Â«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.

248

n.^MENSURATION.

Pyramid and Pyramidic Frustum.â€” 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 â€” J (area of base X

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

X perp height) â€” J <h^.

Volume Vt of pyramidic frustum â€” Vr â€” VÂ« â€”

i(aiAi-aafca). â€ž. ^.

Fig. 24.

If Oa is parallel with Ot. appljring the prismoidal formula, volume Vt of
frustum â€” -~ - 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 Â« \ (perimeter of base X
least slant height) * of a right regular frustum with parallel faces â€” \ (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 â€” 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. â€” 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 coneâ€” J (area of base X perp height) - J athi.
Volume Vo of oblique coneâ€” J (area of base X perp height) â€” J a^.
Volume Vt of conic frustum - V, â€” V. - i (ojfci â€” o^fea) .

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

fnistum â€” . ' 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 facesâ€” 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 â€” ;rr2*. If they are elliptical, use
the formula: area of ellipse â€” ttoo. in which a â€” seim^major axis and 6 â€”
semi-minor axis. Â»r- 3.1416. Digitized by GoOglc

PYRAMID. CONE. WEDGE. SPHERE.

249

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

which Jbj â€” pcrp height. 02 -â– area of elliptic base, and r'â€” Aj ^perpdist

cos ex

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

of obHque cone. Hence, areaÂ»-

oa^a _ 02 cos <X
sin;9

Also, areaâ€” -7 (volume of

oblique cone)â€”

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. â€” If
the wedge is cut from the cone by a plane pass-
ing through the apex (Pig. 26), with Oa parallel to