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   Pyramid & Frustum Formulas   

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also see Defining Geometric Figures

 Pyramid

   A pyramid is a polyhedron of which one side, the base, is a polygon (not necessarily
   a regular polygon), and all the rest are triangles sharing a common point, the vertex.
   A pyramid is regular if the base is a regular polygon and the other faces are congruent
   isosceles triangles
   
Height: h
Area of base: B
Slant height: s (regular pyramid)
Perimeter of base: P
Lateral surface area: S
Volume: V
 

S = sP/2 (regular pyramid)

V = hB/3 

See Ask Dr. Math:
Surface Area of Pyramids
Volume of a Pyramid


 Square Pyramid

   The base is a square, and all triangular faces are congruent isosceles triangles.
 
 
Side of base: a
Other edges: b
Height: h
Slant height: s

Vertex angle of faces: alpha
Base angle of faces: theta
Base-to-face dihedral angle: beta
Face-to-face dihedral angle: phi

Lateral surface area: S
Total surface area (including base): T
Volume: V


a = sqrt[2(b2-h2)] = 2 sqrt(b2-s2) = 2 sqrt(s2-h2)
b = sqrt(h2+a2/2) = sqrt(s2+a2/4) = sqrt(2s2-h2)
h = sqrt(b2-a2/2) = sqrt(s2-a2/4) = sqrt(2s2-b2)
 
  s = sqrt(b2-a2/4) = sqrt(h2+a2/4) = sqrt[(b2+h2)/2]
theta = arccos(a/2b) = arcsin(s/b) = arctan(2s/a)
alpha = arccos(h2/b2) = arcsin(as/b2) = arctan(as/h2)
beta = arccos(a/2s) = arcsin(h/s) = arctan(2h/a)
phi = arccos(-a2/4s2) = arcsin(bh/s2) = arctan(-4bh/a2)

S = 2as
T = a(2s+a)
V = a2h/3


 Frustum of a Pyramid

    The portion of a pyramid that lies between the base and a plane cutting through it
    parallel to the base.
 
Height: h
Area of bases: B1, B2
Slant height: s (regular pyramid)
Perimeter of bases: P1, P2

Lateral surface area: S
Volume: V


     S = s(P1+P2)/2
        (regular pyramid)

     V = h(B1+B2+sqrt[B1B2])/3

 

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