Solids - Volumes and Surfaces
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
d s = 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)
Volume:
Surface:
Parallelepiped
Volume
V = A1 h (3a)
where
A1 = side area (m2, ft2)
Related Sketchup Components from The Engineering ToolBox
Geometric Figures - Cylinders, Boxes, Cones, Planes, Spheres, Lines, Curves and more..
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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)
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 + A b (6b)
where
the surface areas of the triangular faces will have different formulas for different shaped bases
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)
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)
Volume:
Surface:
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)
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