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Pyramid &
Frustum
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Geometric Formulas: Contents 
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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
Basetoface dihedral angle: beta
Facetoface dihedral angle: phi
Lateral surface area: S
Total surface area (including base): T
Volume: V
a = sqrt[2(b^{2}h^{2})] = 2 sqrt(b^{2}s^{2}) =
2 sqrt(s^{2}h^{2})
b = sqrt(h^{2}+a^{2}/2) = sqrt(s^{2}+a^{2}/4) = sqrt(2s^{2}h^{2})
h = sqrt(b^{2}a^{2}/2) = sqrt(s^{2}a^{2}/4) = sqrt(2s^{2}b^{2}) 


s = sqrt(b^{2}a^{2}/4) = sqrt(h^{2}+a^{2}/4) = sqrt[(b^{2}+h^{2})/2]
theta = arccos(a/2b) = arcsin(s/b) = arctan(2s/a)
alpha = arccos(h^{2}/b^{2}) = arcsin(as/b^{2}) = arctan(as/h^{2})
beta = arccos(a/2s) = arcsin(h/s) = arctan(2h/a)
phi = arccos(a^{2}/4s^{2}) = arcsin(bh/s^{2}) = arctan(4bh/a^{2})
S = 2as
T = a(2s+a)
V = a^{2}h/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: B_{1}, B_{2}
Slant height: s (regular pyramid)
Perimeter of bases: P_{1}, P_{2}
Lateral surface area: S
Volume: V
S = s(P_{1}+P_{2})/2
(regular pyramid)
V = h(B_{1}+B_{2}+sqrt[B_{1}B_{2}])/3


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