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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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