Formulas

Elementary Geometry

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Pyramid

  • Side of the base: \(a\)
    Lateral edge: \(b\)
    Height of a pyramid: \(h\)
    Slant height: \(m\)
    Number of sides in the base: \(n\)
    Volume: \(V\)
    Radius of the inscribed circle in the base: \(r\)
    Semiperimeter of the polygon in the base: \(p\)
    Lateral surface area: \({S_L}\)
    Area of the base: \({S_B}\)
    Total surface area: \(S\)
    1. A pyramid is a polyhedron whose base is a polygon and other faces are triangles with a common vertex (also called the apex).
    2. Pyramid
    3. A pyramid whose base is a triangle is called a tetrahedron.
    4. The perpendicular drawn from the vertex of a pyramid to the base plane is called the height (altitude) of the pyramid. In a regular pyramid, the height is given by
      \(h = {\large\frac{{\sqrt {4{b^2}{{\sin }^2}\frac{\pi }{n} – {a^2}} }}{{2\sin \frac{\pi }{n}}}\normalsize},\)
      where \(b\) is the lateral edge, \(a\) is the base side, \(n\) is the number of sides of the polygon in the base.
    5. The height of a lateral face is called the slant height. In a regular pyramid, the length of the slant height is expressed by the formula
      \(m = \sqrt {{b^2} – {\large\frac{{{a^2}}}{4}\normalsize}} \)
    6. Lateral surface area of a regular pyramid
      \({S_L} = {\large\frac{1}{2}\normalsize} man =\) \({\large\frac{1}{4}\normalsize} an\sqrt {4{b^2} – {a^2}} \) \(= pm\)
    7. Area of the base of a regular pyramid
      \({S_B} = pr\),
      where \(p\) is the semiperimeter of the polygon in the base, \(r\) is the inradius of the base.
    8. Total surface area
      \(S = {S_B} + {S_L}\)
    9. Volume of a pyramid
      \(V = {\large\frac{1}{3}\normalsize}{S_B}h\)
    10. Volume of a regular pyramid
      \(V = {\large\frac{1}{3}\normalsize} prh\)