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Solid Geometry Essentials

Solid geometry describes various three-dimensional shapes. There are 5 regular polyhedra including the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Other solids include prisms, pyramids, cylinders, cones, spheres and their variations. Key properties include the number of faces, edges, vertices, and formulas to calculate volume and surface area.

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Rohclem Dela Cruz
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87 views30 pages

Solid Geometry Essentials

Solid geometry describes various three-dimensional shapes. There are 5 regular polyhedra including the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Other solids include prisms, pyramids, cylinders, cones, spheres and their variations. Key properties include the number of faces, edges, vertices, and formulas to calculate volume and surface area.

Uploaded by

Rohclem Dela Cruz
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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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

Octahedron Triangle 8 12 6 4 2𝑎2 3 𝑎3 2


3
Dodecahedron Pentagon 12 30 20 3 20.65𝑎2 7.66𝑎3

Icosahedron Triangle 20 30 12 5 8.66𝑎2 2.18𝑎3

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

𝐹𝑜𝑟 𝑜𝑛𝑒 𝑏𝑎𝑠𝑒:


𝜋ℎ2
𝑉= 3𝑅 − ℎ
3

𝐹𝑜𝑟 𝑡𝑤𝑜 𝑏𝑎𝑠𝑒:


𝜋ℎ
𝑉= (3𝑎2 + 3𝑏 2 + ℎ2 )
6
Spherical Cone

2 2
𝑉 = 𝜋𝑟 ℎ
3
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑍𝑜𝑛𝑒 + 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑜𝑛𝑒
Spherical Pyramid

𝜋𝑟 3 𝐸
𝑉=
540°

Where E is the spherical excess


Torus (Doughnut)

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

Radius of a circumscribed sphere of an Octahedron


𝑎
𝑅= 2
2

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