Volume and Surface of some Common Solids Surface and volume of solids like rectangular prism, cylinder, pyramid, cone and sphere

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Cube


cube

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 31/2  (1c)
where
d = innside diagonal (m, ft)

Cuboid

rectangular prism volume surface area

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 =  (a2 + b2 + c2)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

Volume:  
Surface:  

Parallelepiped

parallelepiped volume surface area

Volume

V = A1 h  (3a)
where
A1 = side area (m2, ft2)

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Cylinder

cylinder volume surface area

Volume

V = π/4 d2 h = π r2 h         (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

Volume:  
Surface:  

Hollow Cylinder

hollow cylinder volume surface area

Volume

V = π/4 h (D2 - d2)   (5)

Pyramid

pyramid volume surface area

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

Volume:  

Frustum of Pyramid

frustum of pyramid volume surface area

Volume

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

Cone

cone volume surface area

Volume

V = 1/3 π r2 h         (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

Volume:  
Surface:  

Side

m = (h2 + r2)1/2   (8c)
A2 / A1 = x2 / h2   (8d)

Frustum of Cone

frustum of cone volume surface area

Volume

V = π/12 h (D2 + D d + d2)   (9a)
m = ( ( (D - d) / 2 )2 + h2)1/2    (9c)

Sphere

sphere volume surface area

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

Volume:  
Surface:  

Zone of a Sphere

zone of a sphere volume surface area
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

segment of a sphere volume surface area
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

sector of a sphere volume surface area
V = 2/3 π r2 h    (13a)
A0 = π/2 r (4 h + s)   (13b)

Sphere with Cylindrical Boring

sphere cylindrical boring volume surface area
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

sphere conical boring volume surface area
V = 2/3 π R2 h   (15a)
A0 = 2 π R (h + (R2 - h2/4)1/2)   (15b)
h = 2 (R2 - r2)1/2    (15c)

Torus

torus volume surface area
V = π2/4 D d2    (16a)
A0 = π2 D d   (16b)

Sliced Cylinder

sliced cylinder volume surface area
V = π/4 d2 h   (17a)
Am = π d h (17b)
A0 = π r (h1 + h2 + r + (r2 + (h1 - h2)2/4)1/2)   (17c)

Ungula

ungula volume surface area
V = 2/3 r2 h   (18a)
Am = 2 r h (18b)
A0 = Am + π/2 r2 + π/2 r (r2 + h2)1/2  (18c)

Barrel

barrel volume surface area
V ≈ π/12 h (2 D2 + d2)   (19a)

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