SOLID GEOMETRY

Euler's Equation for Regular Polyhedron

Faces + Vertices - Edges = 2

Insphere of Polyhedron

A sphere that is inside and is tangent to all faces of the Polyhedron

Midsphere of a Polyhedron

A sphere that is tangent with all edges of the Polyhedron

Circumsphere

A sphere enveloping a Polyhedron with all its vertices tangent to the sphere

Tetra means

Four

Hexa means

Six

Octa means

Eight

Dode (Duode) means

Twelve

Icosa means

Twenty

Total Surface Area of Tetrahedron (4,4,6)

T.S.A. = a²•SQRT(3)

Volume of Tetrahedron (4,4,6)

V = a³•SQRT(2)/12

Altitude of Tetrahedron (4,4,6)

h = a•SQRT(2/3)

Radius of the Circusphere of Tetrahedron (4,4,6)

r = a•SWRT(3/8)

Radius of the Insphere of Tetrahedron (4,4,6)

r = a•SQRT(6)/12

Volume of Tetrahedron with 3 mutually Perpendicular Edges

v = (a + b + c)/6

Total Surface Area of Hexahedron (6, 8, 12)

T.S.A. = 6a²

Volume of Hexahedron (6, 8, 12)

v = a³

Radius of the Insphere of Hexahedron (6, 8, 12)

r = a/2

Radius of the Circumsphere of Hexahedron (6, 8, 12)

r = a•SQRT(3)/2

Total Surface Area of Octahedron (8, 6, 12)

T.S.A. = 2a²•SQRT(3)

Volume of Octahedron (8, 6, 12)

v = a³•SQRT(2)/3

Radius of the Insphere of Octahedron (8, 6, 12)

r = a•SQRT(6)/6

Radius of the Circumsphere of Octahedron (8, 6, 12)

r = a•SQRT(2)/2

Total Surface Area of Dodecahedron (12, 20, 30)

T.S.A. = 20.6457•a²

Volume of Dodecahedron (12, 20, 30)

v = 7.6631•a³

Radius of the Insphere of Dodecahedron (12, 20, 30)

r = 1.1135•a

Radius of the Circumsphere of Dodecahedron (12, 20, 30)

r = 1.4013•a

Total Surface Area of Icosahedron (20, 12, 30)

T.S.A. = 5a²•SQRT(3)

Volume of Icosahedron (20, 12, 30)

v = 2.1817•a³

Radius of the Insphere of Icosahedron (20, 12, 30)

r = 0.7558•a

Radius of the Circumsphere of Icosahedron (20, 12, 30)

r = 0.9511•a

Volume of Any Prism

v = A_base • h

Lateral Area of Any Prism

A = P_base • h

Volume of Cylinder

v = A_base • h

Volume of Cylinder

A = 2πrh

Volume of Any Pyramid

v = ⅓(A_base • h)

Lateral Area of Any Pyramid

A = ½(P_base • L)

Volume of Any Frustum

v = ⅓[A1 + A2 + SQRT(A1A2)] h

Lateral Area of Any Frustum

A = ½(P1 + P2)L

Volume of Cone

v = ⅓A_base •h

Lateral Area of Cone

A = πrL

Volume of Frustum of a Cone

v = ⅓[A1 + A2 + SQRT(A1A2)]h

Volume of a Any Truncated Prism

v = A_base • h_average

Prismatoid

Any solid when cut by parallel plane creates similar shapes

Volume of Prismatoid

v = ⅙(A1 + 4A_mid + A2)h

Volume of Sphere

v = 4/3•πR³

Surface Area of Sphere

A = 4πR²

Area of a Spherical Zone

A = 2πRh

Volume of Spherical Segment

v = ⅓•πh² (3R - h)

Volume of a Spherical Frustum

v = ⅙πh[3(R² + r²) + h²]

Volume of Spherical Sector

v = ⅔πR²h = ⅓A_zone • R

Area of Spherical Polygon

A = πR²E/180

Volume of Spherical Pyramid

v = πR³E/540

Spherical Lune (Area)

A = πR²•θ/90 , θ in degrees

Spherical Wedge (Volume)

v = πR³θ/270 , θ in degrees

Surface Area of Torus

A = 4π²Rr

Volume of Torus

v = 2π²r²R

Volume of a Scalene Ellipsoid

v = 4/3•πabc

Volume of Ellipsoid (Revolved about its Major Axis)

v = 4/3•πab²

Volume of Ellipsoid (Revolved about its Minor Axis)

v = 4/3•πa²b

Volume of Paraboloid

v = ½(πr²h)

Area - Height Relationship of Similar Solids

[A1 / A2] = [h1 / h2]²

Volume - Height Relationship of Similar Solids

[v1 / v2] = [h1 / h2]³

Volume - Area Relationship of Similar Solids

[v1 / v2]² = [A1 / A2]³