Solid Geometry
Solid Geometry
Solid Geometry
Polyhedra
Polyhedra
Plane – is a surface such that a straight line joining any two points lies wholly in
the surface.
Collinear points – are three or more points that lie on the same straight line.
Coplanar points – are points that lie on the same plane.
Angle of Inclination – the angle that the line makes with its projection on a plane
Dihedral Angle – angle formed between two intersecting planes.
Polyhedral Angle – Angle formed by three or more planes which meet at a
common point
Regular Polyhedra
Name Face f e v m Surface Area Volume
Tetrahedron Triangle 4 6 4 3 𝑎2 3 𝑎3
6 2
Hexahedron Square 6 12 8 3 6𝑎2 𝑎3
f = no. of faces, e = no. of edges, v = no. of vertices, m = no. of polygons meeting at a vertex
Solids
Rectangular Parallelepiped
𝑉 = 𝑎𝑏𝑐
d2 c
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 2 𝑎𝑐 + 𝑏𝑐
𝑇𝑜𝑡𝑎𝑙 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2 𝑎𝑐 + 𝑏𝑐 + 𝑎𝑏 d1
𝑑1 = 𝑎2 + 𝑐 2
𝑑2 = 𝑎2 + 𝑏 2 + 𝑐 2 b
a
Cube/ Hexahedron
𝑉 = 𝑎3 d2
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 4𝑎2
𝑇𝑜𝑡𝑎𝑙 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 6𝑎2 d1
𝑑1 = 𝑎 2
𝑑2 = 𝑎 3
a
Prism
Ab
𝑉 = 𝐴𝑏 ℎ
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 𝑃𝑏 ℎ
h
Truncated Prism
h6
𝑉 = 𝐴𝑏𝑎𝑠𝑒 ℎ𝑎𝑣𝑒𝑟𝑎𝑔𝑒
h5
h1 h4
h2 h3
Abase
Cylinder
𝑉 = 𝜋𝑅 2 ℎ h
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 2𝜋𝑅ℎ
𝑇𝑜𝑡𝑎𝑙 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 + 2 ∗ 𝐵𝑎𝑠𝑒 𝐴𝑟𝑒𝑎
Pyramid
1
𝑉 = 𝐴𝑏𝑎𝑠𝑒 ℎ h
3
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 𝑆𝑢𝑚 𝑜𝑓 𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎𝑠
𝑇𝑜𝑡𝑎𝑙 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 + 𝐵𝑎𝑠𝑒 𝐴𝑟𝑒𝑎
Abase
Frustum of a Pyramid
A2
ℎ
𝑉= ( 𝐴1 + 𝐴2 + 𝐴1 𝐴2 )
3
A1
Cone
1 2
𝑉 = 𝜋𝑟 ℎ L
3
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 𝜋𝑟𝐿 h
r
Frustum of a Cone
ℎ
𝑉 = (𝐴1 + 𝐴2 + 𝐴1 𝐴2 )
3
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 = 𝜋(𝑅 + 𝑟)𝐿
Sphere
4
𝑉 = 𝜋𝑟 3
3
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 4𝜋𝑟 2
Spherical Lune
𝜃
𝐴𝑟𝑒𝑎 = 4𝜋𝑟 2
360
Spherical Wedge
4 3 𝜃
𝑉 = 𝜋𝑟
3 360
Spherical Zone
𝐴 = 2𝜋𝑅ℎ
Spherical Segment
2 2
𝑉 = 𝜋𝑟 ℎ
3
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑍𝑜𝑛𝑒 + 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑜𝑛𝑒
Spherical Pyramid
𝜋𝑟 3 𝐸
𝑉=
540°
𝑉 = 2𝜋 2 𝑅𝑟 2
𝑆𝐴 = 4𝜋 2 𝑅𝑟
Ellipsoid
4
𝑉 = 𝜋𝑎𝑏𝑐
3
Oblate Spheroid
4 2
𝑉 = 𝜋𝑏 𝑐
3
Prolate Spheroid
4
𝑉 = 𝜋𝑎𝑐 2
3
Paraboloid
With one base With two bases
1 2 𝜋ℎ 2
𝑉 = 𝜋𝑏 𝑎 𝑉= (𝑅 + 𝑟 2 )
2 2
Hyperboloid
𝜋ℎ 2
𝑉= (𝑅 + 4𝑟 2 )
6
Conoid
𝜋𝑟 2 ℎ
𝑉=
2
h
Prismatoid
ℎ
𝑉 = (𝐴1 + 4𝐴2 + 𝐴3 )
6
Additional
Radius of an inscribed sphere in an Octahedron
𝑎
𝑅= 6
2