Elementary Geometry

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Frustum of Pyramid

Sides of the lower base: \({a_1},{a_2}, \ldots ,{a_n}\)
Sides of the top base: \({b_1},{b_2}, \ldots ,{b_n}\)
Scale factor: \(k\)
Height: \(h\)
Slant height: \(m\)
Perimeters of the bases: \({P_1}\), \({P_2}\)
Areas of the bases: \({S_1}\), \({S_2}\)
Total surface area: \(S\)
Lateral surface area: \({S_L}\)
Volume: \(V\)
  1. A frustum of a pyramid is a polyhedron that lies between the base of the pyramid and a plane through it parallel to the base.
  2. Frustum of Pyramid
  3. The polygons lying in the bases of a frustum of a pyramid are similar to each other:
    \({\large\frac{{{b_1}}}{{{a_1}}}\normalsize} = {\large\frac{{{b_2}}}{{{a_2}}}\normalsize} = {\large\frac{{{b_3}}}{{{a_3}}}\normalsize} = \ldots\) \(= {\large\frac{{{b_n}}}{{{a_n}}}\normalsize} = {\large\frac{b}{a}\normalsize} = k,\)
    where \(k\) is a scale factor.
  4. Ratio of the base areas
    \({\large\frac{{{S_2}}}{{{S_1}}}\normalsize} = {k^2}\)
  5. Lateral surface area of a frustum of a regular pyramid
    \({S_L} = m{\large\frac{{{P_1} + {P_2}}}{2}\normalsize},\)
    where \(m\) is the slant height, \({P_1}\), \({P_2}\) are the perimeters of the top and bottom bases.
  6. Total surface area
    \(S = {S_L} + {S_1} + {S_2}\)
  7. Volume of a frustum of a pyramid
    \(V =\) \({\large\frac{h}{3}\normalsize} \left( {{S_1} + \sqrt {{S_1}{S_2}} + {S_2}} \right)\)