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

 Name Formula Legend Lateral surface area of the cone Show source$S=\pi~{ r}^{2}+\pi~ r~ l$ S - lateral surface area of the cone,r - radius of the cone base,l - slant height of the cone,$\pi$ - pi number (math constant approximately equal to 3.14159). Slant height of the cone Show source$l=\sqrt{{ r}^{2}+{ h}^{2}}$ l - slant height of the cone,r - radius of the cone base,h - height of the cone. Cone volume Show source$V=\frac{1}{3}~\pi~{ r}^{2}~ h$ V - volume of the cone,r - radius of the cone base,h - height of the cone,$\pi$ - pi number (math constant approximately equal to 3.14159).

# Surface areas of solids

 Name Formula Legend Lateral surface area of the cone Show source$S=\pi~{ r}^{2}+\pi~ r~ l$ S - lateral surface area of the cone,r - radius of the cone base,l - slant height of the cone,$\pi$ - pi number (math constant approximately equal to 3.14159). Slant height of the cone Show source$l=\sqrt{{ r}^{2}+{ h}^{2}}$ l - slant height of the cone,r - radius of the cone base,h - height of the cone. Lateral surface area of cylinder Show source$S=2~\pi~{ r}^{2}+2~\pi~ r~ h$ S - lateral surface area of the cylinder,h - height of the cylinder,r - radius of the cylinder base,$\pi$ - pi number (math constant approximately equal to 3.14159). Lateral surface area of the cube Show source$S=6~{ a}^{2}$ S - lateral surface area of the cube,a - cube edge. Total surface area of the cuboid Show source$S=2~ a~ b+2~ a~ h+2~ b~ h$ 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 sphere Show source$S=4~\pi~{ r}^{2}$ S - area of the sphere,r - radius of the sphere,$\pi$ - pi number (math constant approximately equal to 3.14159).

# Volume of solids

 Name Formula Legend Cone volume Show source$V=\frac{1}{3}~\pi~{ r}^{2}~ h$ V - volume of the cone,r - radius of the cone base,h - height of the cone,$\pi$ - pi number (math constant approximately equal to 3.14159). Cylinder volume Show source$V=\pi~{ r}^{2}\cdot h$ V - volume of the cylinder,h - height of the cylinder,r - radius of the cylinder base,$\pi$ - pi number (math constant approximately equal to 3.14159). Cube volume Show source$V={ a}^{3}$ V - cube volume,a - cube edge. Cuboid volume Show source$V= a\cdot b\cdot h$ 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 volume Show source$V=\frac{4}{3}~\pi~{ r}^{3}$ V - volume of the sphere,r - radius of the sphere,$\pi$ - pi number (math constant approximately equal to 3.14159).

# Cylinder

 Name Formula Legend Lateral surface area of cylinder Show source$S=2~\pi~{ r}^{2}+2~\pi~ r~ h$ S - lateral surface area of the cylinder,h - height of the cylinder,r - radius of the cylinder base,$\pi$ - pi number (math constant approximately equal to 3.14159). Cylinder volume Show source$V=\pi~{ r}^{2}\cdot h$ V - volume of the cylinder,h - height of the cylinder,r - radius of the cylinder base,$\pi$ - pi number (math constant approximately equal to 3.14159).

# Prisms

 Name Formula Legend Lateral surface area of the cube Show source$S=6~{ a}^{2}$ S - lateral surface area of the cube,a - cube edge. Cube volume Show source$V={ a}^{3}$ V - cube volume,a - cube edge. Total surface area of the cuboid Show source$S=2~ a~ b+2~ a~ h+2~ b~ h$ 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 volume Show source$V= a\cdot b\cdot h$ 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

 Name Formula Legend Lateral surface area of the cube Show source$S=6~{ a}^{2}$ S - lateral surface area of the cube,a - cube edge. Cube volume Show source$V={ a}^{3}$ V - cube volume,a - cube edge.

# Cuboid

 Name Formula Legend Total surface area of the cuboid Show source$S=2~ a~ b+2~ a~ h+2~ b~ h$ 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 volume Show source$V= a\cdot b\cdot h$ 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

 Name Formula Legend Lateral surface area of the sphere Show source$S=4~\pi~{ r}^{2}$ S - area of the sphere,r - radius of the sphere,$\pi$ - pi number (math constant approximately equal to 3.14159). Sphere volume Show source$V=\frac{4}{3}~\pi~{ r}^{3}$ V - volume of the sphere,r - radius of the sphere,$\pi$ - 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:
$S = 6a^2$
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:
$V = a^3$
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 ($m^2$).
• 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 ($dm^3$).
• 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).