Mathematical tables: typical stereometry related formulas
Table shows common formulas related to shapes (solids) in three dimensions i.e. so-called stereometry.

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Cone

NameFormulaLegend
Lateral surface area of the coneShow sourceS=π r2+π r l S=\pi~{ r}^{2}+\pi~ r~ l
  • S - lateral surface area of the cone,
  • r - radius of the cone base,
  • l - slant height of the cone,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Slant height of the coneShow sourcel=r2+h2 l=\sqrt{{ r}^{2}+{ h}^{2}}
  • l - slant height of the cone,
  • r - radius of the cone base,
  • h - height of the cone.
Cone volumeShow sourceV=13 π r2 h V=\frac{1}{3}~\pi~{ r}^{2}~ h
  • V - volume of the cone,
  • r - radius of the cone base,
  • h - height of the cone,
  • π\pi - pi number (math constant approximately equal to 3.14159).

Surface areas of solids

NameFormulaLegend
Lateral surface area of the coneShow sourceS=π r2+π r l S=\pi~{ r}^{2}+\pi~ r~ l
  • S - lateral surface area of the cone,
  • r - radius of the cone base,
  • l - slant height of the cone,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Slant height of the coneShow sourcel=r2+h2 l=\sqrt{{ r}^{2}+{ h}^{2}}
  • l - slant height of the cone,
  • r - radius of the cone base,
  • h - height of the cone.
Lateral surface area of cylinderShow sourceS=2 π r2+2 π r h S=2~\pi~{ r}^{2}+2~\pi~ r~ h
  • S - lateral surface area of the cylinder,
  • h - height of the cylinder,
  • r - radius of the cylinder base,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Lateral surface area of the cubeShow sourceS=6 a2 S=6~{ a}^{2}
  • S - lateral surface area of the cube,
  • a - cube edge.
Total surface area of the cuboidShow sourceS=2 a b+2 a h+2 b h S=2~ a~ b+2~ a~ h+2~ b~ h
  • S - lateral surface area of the cuboid,
  • a - first base edge of the cuboid,
  • b - second base edge of the cuboid,
  • h - height of the cuboid.
Lateral surface area of the sphereShow sourceS=4 π r2 S=4~\pi~{ r}^{2}
  • S - area of the sphere,
  • r - radius of the sphere,
  • π\pi - pi number (math constant approximately equal to 3.14159).

Volume of solids

NameFormulaLegend
Cone volumeShow sourceV=13 π r2 h V=\frac{1}{3}~\pi~{ r}^{2}~ h
  • V - volume of the cone,
  • r - radius of the cone base,
  • h - height of the cone,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Cylinder volumeShow sourceV=π r2h V=\pi~{ r}^{2}\cdot h
  • V - volume of the cylinder,
  • h - height of the cylinder,
  • r - radius of the cylinder base,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Cube volumeShow sourceV=a3 V={ a}^{3}
  • V - cube volume,
  • a - cube edge.
Cuboid volumeShow sourceV=abh V= a\cdot b\cdot h
  • V - volume of the cuboid,
  • a - first base edge of the cuboid,
  • b - second base edge of the cuboid,
  • h - height of the cuboid.
Sphere volumeShow sourceV=43 π r3 V=\frac{4}{3}~\pi~{ r}^{3}
  • V - volume of the sphere,
  • r - radius of the sphere,
  • π\pi - pi number (math constant approximately equal to 3.14159).

Cylinder

NameFormulaLegend
Lateral surface area of cylinderShow sourceS=2 π r2+2 π r h S=2~\pi~{ r}^{2}+2~\pi~ r~ h
  • S - lateral surface area of the cylinder,
  • h - height of the cylinder,
  • r - radius of the cylinder base,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Cylinder volumeShow sourceV=π r2h V=\pi~{ r}^{2}\cdot h
  • V - volume of the cylinder,
  • h - height of the cylinder,
  • r - radius of the cylinder base,
  • π\pi - pi number (math constant approximately equal to 3.14159).

Prisms

NameFormulaLegend
Lateral surface area of the cubeShow sourceS=6 a2 S=6~{ a}^{2}
  • S - lateral surface area of the cube,
  • a - cube edge.
Cube volumeShow sourceV=a3 V={ a}^{3}
  • V - cube volume,
  • a - cube edge.
Total surface area of the cuboidShow sourceS=2 a b+2 a h+2 b h S=2~ a~ b+2~ a~ h+2~ b~ h
  • S - lateral surface area of the cuboid,
  • a - first base edge of the cuboid,
  • b - second base edge of the cuboid,
  • h - height of the cuboid.
Cuboid volumeShow sourceV=abh V= a\cdot b\cdot h
  • V - volume of the cuboid,
  • a - first base edge of the cuboid,
  • b - second base edge of the cuboid,
  • h - height of the cuboid.

Cube

NameFormulaLegend
Lateral surface area of the cubeShow sourceS=6 a2 S=6~{ a}^{2}
  • S - lateral surface area of the cube,
  • a - cube edge.
Cube volumeShow sourceV=a3 V={ a}^{3}
  • V - cube volume,
  • a - cube edge.

Cuboid

NameFormulaLegend
Total surface area of the cuboidShow sourceS=2 a b+2 a h+2 b h S=2~ a~ b+2~ a~ h+2~ b~ h
  • S - lateral surface area of the cuboid,
  • a - first base edge of the cuboid,
  • b - second base edge of the cuboid,
  • h - height of the cuboid.
Cuboid volumeShow sourceV=abh V= a\cdot b\cdot h
  • V - volume of the cuboid,
  • a - first base edge of the cuboid,
  • b - second base edge of the cuboid,
  • h - height of the cuboid.

Sphere

NameFormulaLegend
Lateral surface area of the sphereShow sourceS=4 π r2 S=4~\pi~{ r}^{2}
  • S - area of the sphere,
  • r - radius of the sphere,
  • π\pi - pi number (math constant approximately equal to 3.14159).
Sphere volumeShow sourceV=43 π r3 V=\frac{4}{3}~\pi~{ r}^{3}
  • V - volume of the sphere,
  • r - radius of the sphere,
  • π\pi - pi number (math constant approximately equal to 3.14159).

Some facts

  • Stereometry is a branch of mathematics dealing with the study of three-dimensional solids and the relationships between them.
  • Stereometry is the equivalent of a plane, two-dimensional geometry (sometimes called planimetry) in three-dimensional space. For this reason the term solid geometry, three-dimensional geometry or 3D geometry are used.
  • Spatial shapes (equivalents of flat figures in three-dimensional space) are often called solids. Examples of typical solids are, among others:
    • sphere - it is a generalization of the circle into three-dimensional space,
    • cuboid - generalization of rectangle,
    • cube - generalization of square,
    • cone,
    • cylinder,
    • itd.
  • The most typical solid properties are:
    • lateral surface area - it is the sum of all external surfaces of the solid, e.g. in case of a cube, there are six identical walls with the same area, therefore the cube field is:
      S=6a2S = 6a^2
      where:
      • S - lateral surface area of the cube,
      • a - the length of the cube edge, it is equal between all the edges: the width of the base, the length of the base and the height of the whole cube.
    • volume - determines how much space the given solid occupies, e.g. the cube volume is:
      V=a3V = a^3
      where:
      • V - cube volume,
      • a - the length of the cube's edge.
  • The lateral surface area is the sum of the flat figures. When measuring the total area, we always deal with square units, for example, square meters (m2 m^2).
  • You can find more about square units and the concept of surface area in our other calculator: Area.
  • If you want to learn more about the geometry of flat figures (2D), check out our other calculator: Math tables: geometry.
  • The volume of the solid is always given in cubic units e.g. cubic decimeter (dm3dm^3).
  • Volume is a property unique to spatial solids. It means that it has no sense for flat, two-dimensional shapes.
  • If you want to learn more about cubic units and the volume, check out our other calculator Volume (capacity).

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