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 $S=\pi \cdot r^{2}+\pi \cdot r \cdot l$	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 $V=\frac{1}{3}~\pi \cdot r^{2} \cdot h$	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 $S=\pi \cdot r^{2}+\pi \cdot r \cdot l$	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 $S=2~\pi \cdot r^{2}+2~\pi \cdot r \cdot h$	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 $S=2~a \cdot b+2~a \cdot h+2~b \cdot h$	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 $S=4~\pi \cdot r^{2}$	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 $V=\frac{1}{3}~\pi \cdot r^{2} \cdot h$	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 $V=\pi \cdot r^{2} \cdot h$	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 $V=\frac{4}{3}~\pi \cdot r^{3}$	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 $S=2~\pi \cdot r^{2}+2~\pi \cdot r \cdot h$	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 $V=\pi \cdot r^{2} \cdot h$	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 $S=2~a \cdot b+2~a \cdot h+2~b \cdot h$	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 $S=2~a \cdot b+2~a \cdot h+2~b \cdot h$	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 $S=4~\pi \cdot r^{2}$	S - area of the sphere, r - radius of the sphere, $\pi$ - pi number (math constant approximately equal to 3.14159).
Sphere volume	Show source $V=\frac{4}{3}~\pi \cdot r^{3}$	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 units.
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) units.

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