A = area one end;

a = area of other end;

m = area of section midway between ends; 7 = perpendicular distance between ends. V = (A + a + 4 m).

The area m is not in general a mean between the areas of the two ends, but its sides are means between the corresponding lengths of the ends.

Approximately,

V

A

3

To obtain area of base, divide it into triangles, and find

their sum.

FRUSTUM

A+ a
2

REGULAR PYRAMID.

P= perimeter of base;
A area of base.

C = Pl.
SPIA.

A h

The formula for V applies to any pyramid whose base is A and altitude h.

(D+d).

2

π

= 7 ( R2 — r2.

1.

a

A

V =

OF REGULAR PYRAMID.
area of upper base;

area of lower base;

V = } h ( A + a + Vдa).

The formula for V applies to the frustum of any pyramid..

p

perimeter of upper base;

P = perimeter of lower base.

C = { 1 (P + p).

S = 1 (P + p) + A + a.

LENGTH OF SPIRAL.

n = number of coil;
l= length of spiral;

t = pitch.

PRISM OR PARALLELOPIPED.

C= Ph.

S Ph+2 A.
V = A h.

For prisms with regular polygon as
bases, P= length of one side X number of sides.

To obtain area of base, if it is a polygon, divide it into triangles, and find sum of partial areas.

V = section.

FRUSTUM OF PRISM.

If a section perpendicular to the edges is a triangle, square, parallelogram, or regular polygon, sum of lengths of edges X area of right number of edges

REGULAR POLYGONS.

Divide the polygon into equal triangles and find the sum of the partial areas. Otherwise, square the length of one side and multiply by proper number from the following table: Name. No. Sides. Multiplier Triangle

Square
Pentagon

D

Hexagon
Heptagon
Octagon
Nonagon
Decagon

IRREGULAR AREAS.

Divide the area into trapezoids, triangles, parts of circles, etc., and find the sum of the partial areas. If the figure is very irregular, the approximate area may be found as follows: Divide the figure into parts by equidistant parallel lines b, c, d, etc. The lengths of these lines being measured, then, calling a the first and n the last length, and y the width of strips,

Area

a + n
2

= y

3

4

5

6

7

8

9

10

.433

1.000

1.720

2.598

3.634

4.828

6.182

7.694

m).

+b+c+ etc. + m

CIRCUMFERENCES AND AREAS OF CIRCLES FROM 1-64 TO 100.

.0491

.0982

.1963

.3927

.5890

.7854

.9817

1.1781

1.3744

1.5708

1.7671

1.9635

2.1598

2.3562

2.5525

2.7489

2.9452

3.1416

3.5343

3.9270

4.3197

4.7124

5.1051

.0002

.0008

.0031

.0123

.0276

.0491

.0767

.1104

.1503

.1963

.2485

.3068

.3712

.4418

.5185

.6013

.6903

.7854

.9940

1.2272

1.4849

1.7671

2.0739

5.4978

2.4053

5.8905

2.7612

6.2832

3.1416

6.6759 3.5466

7.0686

3.9761

7.4613

4.4301

7.8540

4.9087
5.4119

8.2467
8.6394

5.9396

6.4918

9.0321
9.4248

7.0686

9.8175

7.6699

10.2102

8.2958

10.6029
10.9956

8.9462
9.6211
11.3883 10.3206
11.7810 11.0447
12.1737 11.7933
12.5664 12.5664

12.9591 13.3641

13.3518

14.1863

Circum. Area.

13.7445

14.1372

14.5299

14.9226

15.3153

15.7080

16.1007

16.4934

16.8861

17.2788

17.6715

18.0642

18.4569

18.8496

19.2423

19.6350

20.0277

20.4204

20.8131

21.2058

21.5985
21.9912

22.3839

22.7766

23.1693
23.5620

23.9547

24.3474
24.7401
25.1328
25.5255

25.9182

26.3109

26.7036

27.0963

27.4890

27.8817
28.2744
28.6671

29.0598

29.4525

29.8452
30.2379

30.6306

15.0330

15.9043

16.8002

17.7206

18.6555

19.6350

20.6290

21.6476

22.6907

23.7583

24.8505

25.9673

27.1086

28.2744

29.4648

30.6797

31.9191

33.1831

34.4717

35.7848

37.1224

38.4846

39.8713

41.2826

42.7184

44.1787

45.6636

47.1731

48.7071

50.2656

51.8487

53.4563

55.0884

56.7451

58.4264

60.1322

61.8625

63.6174

65.3968

67.2008

69.0293

70.8823

72.7599

74.6621

TABLE (Continued).

Circum. Area. Diam. Circum. Area.

15

151

15

16

161

16

161

16

16

161

167

17

17층

17

171

174

17

171

171

18

181

18!

181

18

18

181

187

19

191

19

19

19

19

193

19급

20

201

201

203

201

201

21

211

211