Solids - Volumes And Surfaces - The Engineering ToolBox

Cube

• all figures made with SketchUp and the Engineering ToolBox SketchUp Extension

Volume

V = a3 (1)

where

V = volume (m3, ft3 )

a = side (m, ft)

Surface Area

A0 = 6 a2(1b)

where

A0 = surface area (m2, ft2)

Diagonal

d = a 3 1/2 (1c)

where

d = inside diagonal (m, ft)

Diagonal of Cube Face

ds = a 2 1/2 (1d)

Cuboid - Square Prism

Volume

V = a b c (2)

where

V = volume of solid (m3, ft3 )

a = length of rectangular prism (m, ft)

b = width of rectangular prism (m, ft)

c = height of rectangular prism (m, ft)

Diagonal

d = (a 2 + b 2 + c 2 ) 1/2 (2b)

Surface Area

A0 = 2 (a b + a c + b c) (2c)

where

A0 = surface area of solid (m2, ft2)

length

width

height

m dm cm mm in ft

Volume:

Surface:

Parallelepiped

Volume

V = A1 h (3a)

where

A1 = side area (m2, ft2)

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Cylinder

Volume

V = π r2h = ( π / 4) d2h (4a)

where

d = diameter of cylinder (m, ft)

r = radius of cylinder (m, ft)

h = height of cylinder (m, ft)

Surface

A = 2 π r h + 2 π r2(4b)

radius

height

m dm cm mm in ft

Volume:

Surface:

Hollow Cylinder

Volume

V = π/4 h (D2- d2) (5)

Pyramid

Volume

V = 1/3 h A1 (6)

where

A1 = area of base (m2, ft2)

h = perpendicular height of pyramid (m, ft)

Surface

A = ∑ sum of areas of triangles forming sides + Ab (6b)

where

the surface areas of the triangular faces will have different formulas for different shaped bases

area of base

perpendicular height

m dm cm mm in ft

Volume:

Frustum of Pyramid

Volume

V = h/3 ( A1 + A2+ (A1 A2) 1/2 ) (7)

Cone

Volume

V = 1/3 π r2h (8a)

where

r = radius of cone base (m, ft)

h = height of cone (m, ft)

Surface

A = π r l + π r2(8b)

where

l = (r2+ h2) 1/2 = length of cone side (m, ft)

radius

height

m dm cm mm in ft

Volume:

Surface:

Side

m = (h2+ r2) 1/2 (8c)

A2/ A1 = x2/ h2(8d)

Frustum of Cone

Volume

V = π/12 h (D2+ D d + d2) (9a)

m = ( ( (D - d) / 2 )2+ h2) 1/2 (9c)

Sphere

Volume

V = 4/3 π r3

= 1/6 π d3 (10a)

where

r = radius of sphere (m, ft)

Surface

A = 4 π r2

= π d2 (10b)

radius

m dm cm mm in ft

Volume:

Surface:

Spheres with Fractional Diameters - Surface Areas and Volumes

Spheres with Fractional Diameters - Surface Areas and Volumes

Fraction Diameter - d – (inch)	Decimal Diameter - d – (inch)	Decimal Radius – r – (inch)	Surface Area - A – (in2)	Volume - V - (in3 )
1/64	0.015625	0.007813	0.0007670	0.0000020
1/32	0.031250	0.015625	0.0030680	0.0000160
3/64	0.046875	0.023438	0.0069029	0.0000539
1/64	0.062500	0.031250	0.0122718	0.0001278
5/64	0.078125	0.039063	0.0191748	0.0002497
3/32	0.093750	0.046875	0.0276117	0.0004314
7/64	0.109375	0.054688	0.0375825	0.0006851
1/8	0.125000	0.062500	0.0490874	0.0010227
9/64	0.140625	0.070313	0.0621262	0.0014561
5/32	0.156250	0.078125	0.0766990	0.0019974
11/64	0.171875	0.085938	0.0928058	0.0026585
3/16	0.187500	0.093750	0.1104466	0.0034515
13/64	0.203125	0.101563	0.1296214	0.0043882
7/32	0.218750	0.109375	0.1503301	0.0054808
15/64	0.234375	0.117188	0.1725728	0.0067411
1/4	0.250000	0.125000	0.1963495	0.0081812
17/64	0.265625	0.132813	0.2216602	0.0098131
9/32	0.281250	0.140625	0.2485049	0.0116487
19/64	0.296875	0.148438	0.2768835	0.0137000
5/16	0.312500	0.156250	0.3067962	0.0159790
21/64	0.328125	0.164063	0.3382428	0.0184977
11/32	0.343750	0.171875	0.3712234	0.0212680
23/64	0.359375	0.179688	0.4057379	0.0243020
3/8	0.375000	0.187500	0.4417865	0.0276117
25/64	0.390625	0.195313	0.4793690	0.0312089
13/32	0.406250	0.203125	0.5184855	0.0351058
27/64	0.421875	0.210938	0.5591360	0.0393142
7/16	0.437500	0.218750	0.6013205	0.0438463
29/64	0.453125	0.226563	0.6450389	0.0487139
15/32	0.468750	0.234375	0.6902914	0.0539290
31/64	0.484375	0.242188	0.7370778	0.0595037
1/2	0.500000	0.250000	0.7853982	0.0654498
33/64	0.515625	0.257813	0.8352525	0.0717795
17/32	0.531250	0.265625	0.8866409	0.0785047
35/64	0.546875	0.273438	0.9395632	0.0856373
9/16	0.562500	0.281250	0.9940196	0.0931893
37/64	0.578125	0.289063	1.0500098	0.1011728
19/32	0.593750	0.296875	1.1075341	0.1095997
39/64	0.609375	0.304688	1.1665924	0.1184820
5/8	0.625000	0.312500	1.2271846	0.1278317
41/64	0.640625	0.320313	1.2893109	0.1376608
21/32	0.656250	0.328125	1.3529711	0.1479812
43/64	0.671875	0.335938	1.4181652	0.1588050
11/16	0.687500	0.343750	1.4848934	0.1701440
45/64	0.703125	0.351563	1.5531555	0.1820104
23/32	0.718750	0.359375	1.6229517	0.1944161
47/64	0.734375	0.367188	1.6942818	0.2073730
3/4	0.750000	0.375000	1.7671459	0.2208932
49/64	0.765625	0.382813	1.8415439	0.2349887
25/32	0.781250	0.390625	1.9174760	0.2496714
51/64	0.796875	0.398438	1.9949420	0.2649532
13/16	0.812500	0.406250	2.0739420	0.2808463
53/64	0.828125	0.414063	2.1544760	0.2973626
27/32	0.843750	0.421875	2.2365440	0.3145140
55/64	0.859375	0.429688	2.3201459	0.3323126
7/8	0.875000	0.437500	2.4052819	0.3507703
57/64	0.890625	0.445313	2.4919518	0.3698991
29/32	0.906250	0.453125	2.5801557	0.3897110
59/64	0.921875	0.460938	2.6698936	0.4102180
15/16	0.937500	0.468750	2.7611654	0.4314321
61/64	0.953125	0.476563	2.8539713	0.4533652
31/32	0.968750	0.484375	2.9483111	0.4760294
63/64	0.984375	0.492188	3.0441849	0.4994366
1	1.000000	0.500000	3.1415927	0.5235988

Zone of a Sphere

V = π/6 h (3a2+ 3b2+ h) (11a)

Am = 2 π r h (11b)

A0 = π (2 r h + a2+ b2) (11c)

Segment of a Sphere

V = π/6 h (3/4 s2+ h2)

= π h2(r - h/3) (12a)

Am = 2 π r h

= π/4 (s2+ 4 h2) (12b)

Sector of a Sphere

V = 2/3 π r2h (13a)

A0 = π/2 r (4 h + s) (13b)

Sphere with Cylindrical Boring

V = π/6 h3 (14a)

A0 = 4 π ((R + r)3 (R - r)) 1/2

= 2 π h (R + r) (14b)

h = 2 (R2- r2) 1/2 (14c)

Sphere with Conical Boring

V = 2/3 π R2h (15a)

A0 = 2 π R (h + (R2- h2/4) 1/2 ) (15b)

h = 2 (R2- r2) 1/2 (15c)

Torus

V = π2/4 D d2 (16a)

A0 = π2D d (16b)

Sliced Cylinder

V = π/4 d2h

= π r2((h1 + h2) / 2) (17a)

Am = π d h

= 2 π r ((h1 + h2) / 2) (17b)

where

Am = side walls area

A0 = π r (h1 + h2+ r + (r2+ (h1 - h2)2/4) 1/2 ) (17c)

where

A0 = surface area

Ungula

V = 2/3 r2h (18a)

Am = 2 r h (18b)

A0 = Am + π/2 r2+ π/2 r (r2+ h2) 1/2 (18c)

Barrel

V ≈ π/12 h (2 D2+ d2) (19a)