Formulas and Tables

Elementary Geometry

Sphere

Radius of a sphere: \(R\)
Height of a spherical cap/segment: \(h\)
Radius of the base of a spherical cap: \(r\)
Area of the base of a spherical cap: \({S_B}\)
Area of the spherical surface of a cap: \({S_C}\)

Radii of the bases of a spherical segment: \({r_1},\) \({r_2}\)
Areas of the bases of a spherical segment: \({S_1},\) \({S_2}\)
Area of the spherical surface of a segment: \({S_S}\)
Total surface area \(S\)
Volume: \(V\)

  1. A sphere is the set of all points in a space equidistant from a given point called the center of the sphere. The distance between any point of the sphere and its centre is called the radius. The inside of a sphere is called a ball.
Sphere
  1. Surface area of a sphere
    \(S = 4\pi {R^2}\)
  2. Volume of a sphere
    \(V = {\large\frac{{4\pi {R^3}}}{3}\normalsize}\)
  3. A spherical cap is a portion of a sphere cut off by a plane.
A spherical cap
  1. Relationship between the height and base radius of a spherical cap
    \(R = {\large\frac{{{r^2} + {h^2}}}{{2h}}\normalsize},\)
    where \(h\) is the height of the spherical cap, \(r\) is the base radius of the spherical cap, \(R\) is the radius of the sphere.
  2. Base area of a spherical cap
    \({S_B} = \pi {r^2}\)
  3. Area of the spherical surface of a spherical cap
    \({S_C} = \pi \left( {{h^2} + {r^2}} \right) \)
  4. Total surface area of a spherical cap
    \(S = {S_B} + {S_C} =\) \(\pi \left( {{h^2} + 2{r^2}} \right) \) \(= \pi \left( {2Rh + {r^2}} \right) \)
  5. Volume of a spherical cap
    \(V = {\large\frac{{\pi {h^2}\left( {3R – h} \right)}}{6}\normalsize} \) \(= {\large\frac{{\pi h\left( {3{r^2} + {h^2}} \right)}}{6}\normalsize}\)
  6. A spherical segment is a portion of a sphere that lies between two parallel planes cutting the sphere.
A spherical segment
  1. Area of the spherical surface of a spherical segment
    \({S_S} = 2\pi Rh\),
    where \(h\) is the height of the spherical segment, \(R\) is the radius of the sphere.
  2. Total surface area of a spherical segment
    \(S = {S_S} + {S_1} + {S_2} \) \(= \pi \left( {2Rh + r_1^2 + r_2^2} \right),\)
    where \(h\) is the height of the spherical segment, \(R\) is the radius of the sphere, \({r_1}\), \({r_2}\) are the radii of the bases of the segment, \({S_1}\), \({S_2}\) are the areas of these bases.
  3. Volume of a spherical segment
    \(V =\) \( {\large\frac{{\pi h\left( {3r_1^2 + 3r_2^2 + {h^2}} \right)}}{6}\normalsize},\)
    where \({r_1}\), \({r_2}\) are the radii of the bases of the spherical segment, \(h\) is its height.
  4. A spherical sector is a portion of a sphere consisting of a spherical cap and the cone with vertex at the centre of the sphere and the base of the spherical cap. It is assumed here that the spherical cap is less that the hemisphere.
A spherical sector
  1. Total surface area of a spherical sector
    \(S = \pi R\left( {2h + r} \right),\)
    where \(h\) is the height of the corresponding spherical cap, \(r\) is the base radius of the cap (or the cone), \(R\) is the radius of the sphere.
  2. Volume of a spherical sector
    \(V = {\large\frac{{2\pi {R^2}h}}{3}\normalsize}\)