Elementary Geometry

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Lateral edge: \(\ell\)
Altitude: \(h\)
Sides of the base: \({a_1},{a_2}, \ldots ,{a_n}\)
Perimeter of the cross section: \(p\)
Volume: \(V\)
Lateral surface area: \({S_L}\)
Area of the base: \({S_B}\)
Area of the perpendicular cross section: \({S_C}\)
Total surface area: \(S\)
  1. A prism is a polyhedron whose bases are polygons and the lateral faces are parallelograms. The bases of a prism are equal polygons lying in parallel planes.
  2. Prism
  3. A prism is called a right prism if its lateral edges are perpendicular to the bases. Otherwise it is an oblique prism.
  4. If the bases of a prism are parallelograms, then the prism is called a parallelepiped. In a particular case, when the bases are rectangles and the prism is a right prism, it is called a rectangular parallelepiped.
  5. A right prism is called regular if its bases are regular polygons. In particular, if the bases and lateral faces of a prism are squares, the prism is called a cube.
  6. Lateral surface area of a reqular prism
    \({S_L} = {P_B}\ell =\) \( \left( {{a_1} + {a_2} + \ldots + {a_n}} \right)\ell,\)
    where \({P_B}\) is the perimeter of the base of the prism, \({a_1},\) \({a_2}, \ldots ,\) \({a_n}\) are the sides of the base, \(\ell\) is the length of the lateral edge (in a right prism, the lateral edge coincides with the altitude \(h\)).
  7. Lateral surface area of an oblique prism
    \({S_L} = p\ell\),
    where \(p\) is the semiperimeter of a cross section of the prism, \(\ell\) is the lateral edge.
  8. Volume of a prism
    \(V = {S_B}h = {S_C}\ell\),
    where \({S_B}\) is the base area, \(h\) is the altitude of the prism, \({S_C}\) is the area of a cross section, \(\ell\) is the lateral edge.
  9. Cavalieri’s principle
    Given two solids included between parallel planes. If every plane cross section parallel to the given planes has the same area in both solids, then the volumes of the solids are equal.