List of formulas in elementary geometry

Short description: none

This is a short list of some common mathematical shapes and figures and the formulas that describe them.

## Contents

* 1 Two-dimensional shapes
* 2 Three-dimensional shapes
* 2.1 Sphere
* 3 See also
* 4 References

## Two-dimensional shapes

Shape | Area | Perimeter/Circumference | Meanings of symbols

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Square | [math]\displaystyle{ l^2 }[/math] | [math]\displaystyle{ 4l }[/math] | [math]\displaystyle{ l }[/math] is the length of a side
Rectangle | [math]\displaystyle{ lb }[/math] | [math]\displaystyle{ 2(l+b) }[/math] | [math]\displaystyle{ l }[/math] is length, [math]\displaystyle{ b }[/math] is breadth
Circle | [math]\displaystyle{ \pi r^2 }[/math] | [math]\displaystyle{ 2\pi r }[/math] or [math]\displaystyle{ \pi d }[/math] | where [math]\displaystyle{ r }[/math] is the radius and [math]\displaystyle{ d }[/math] is the diameter
Ellipse | [math]\displaystyle{ \pi ab }[/math] |  | where [math]\displaystyle{ a }[/math] is the semimajor axis and [math]\displaystyle{ b }[/math] is the semiminor axis
Triangle | [math]\displaystyle{ \frac{bh}{2} }[/math] | [math]\displaystyle{ a+b+c }[/math] | [math]\displaystyle{ b }[/math] is base; [math]\displaystyle{ h }[/math] is height; [math]\displaystyle{ a,b,c }[/math] are sides
Parallelogram | [math]\displaystyle{ bh }[/math] | [math]\displaystyle{ 2(a+b) }[/math] | [math]\displaystyle{ b }[/math] is base, [math]\displaystyle{ h }[/math] is height, [math]\displaystyle{ a }[/math] is side
Trapezoid | [math]\displaystyle{ \frac{a+b}{2}h }[/math] |  | [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are the bases

Sources:[1][2][3]

## Three-dimensional shapes

Illustration of the shapes' equation terms

Cube

Cuboid

Prism

Parallelepiped

Pyramids

Tetrahedron

Cone

Cylinder

Sphere

Ellipsoid

This is a list of volume formulas of basic shapes:[4](pp405–406)

* Cone – [math]\displaystyle{ \frac{1}{3}\pi r^2 h }[/math], where [math]\displaystyle{ r }[/math] is the base's radius
* Cube – [math]\displaystyle{ a^3 }[/math], where [math]\displaystyle{ a }[/math] is the side's length;
* Cuboid – [math]\displaystyle{ abc }[/math], where [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], and [math]\displaystyle{ c }[/math] are the sides' length;
* Cylinder – [math]\displaystyle{ \pi r^2 h }[/math], where [math]\displaystyle{ r }[/math] is the base's radius and [math]\displaystyle{ h }[/math] is the cone's height;
* Ellipsoid – [math]\displaystyle{ \frac{4}{3}\pi abc }[/math], where [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], and [math]\displaystyle{ c }[/math] are the semi-major and semi-minor axes' length;
* Sphere – [math]\displaystyle{ \frac{4}{3}\pi r^3 }[/math], where [math]\displaystyle{ r }[/math] is the radius;
* Parallelepiped – [math]\displaystyle{ abc\sqrt{K} }[/math], where [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], and [math]\displaystyle{ c }[/math] are the sides' length,[math]\displaystyle{ K = 1 + 2\cos(\alpha)\cos(\beta)\cos(\gamma) - \cos^2(\alpha) - \cos^2(\beta) - \cos^2(\gamma) }[/math], and [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math], and [math]\displaystyle{ \gamma }[/math] are angles between the two sides;
* Prism – [math]\displaystyle{ Bh }[/math], where [math]\displaystyle{ B }[/math] is the base's area and [math]\displaystyle{ h }[/math] is the prism's height;
* Pyramid – [math]\displaystyle{ \frac{1}{3}Bh }[/math], where [math]\displaystyle{ B }[/math] is the base's area and [math]\displaystyle{ h }[/math] is the pyramid's height;
* Tetrahedron – [math]\displaystyle{ {\sqrt{2}\over12}a^3 }[/math], where [math]\displaystyle{ a }[/math] is the side's length.

### Sphere

The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables

* [math]\displaystyle{ r }[/math] is the radius,
* [math]\displaystyle{ C = 2 \pi r }[/math] is the circumference (the length of any one of its great circles),
* [math]\displaystyle{ S }[/math] is the surface area,
* [math]\displaystyle{ V }[/math] is the volume.

Surface area:

[math]\displaystyle{ \begin{alignat}{4} S &= 4 \pi r^2 \\\\[0.3ex] &= \frac{1}{\pi} C^2 \\\\[0.3ex] &= \sqrt[3]{\pi (6 V)^2} \\\\[0.3ex] \end{alignat} }[/math]

Volume:

[math]\displaystyle{ \begin{alignat}{4} V &= \frac{4}{3} \pi r^3 \\\\[0.3ex] &= \frac{1}{6 \pi^2} C^3 \\\\[0.3ex] &= \frac{1}{6 \sqrt{\pi}} S^{3/2} \\\\[0.3ex] \end{alignat} }[/math]

Radius:

[math]\displaystyle{ \begin{alignat}{4} r &= \frac{1}{2 \pi} C \\\\[0.3ex] &= \sqrt{\frac{1}{4 \pi} S} \\\\[0.3ex] &= \sqrt[3]{\frac{3}{4 \pi} V} \\\\[0.3ex] \end{alignat} }[/math]

Circumference:

[math]\displaystyle{ \begin{alignat}{4} C &= 2 \pi r \\\\[0.3ex] &= \sqrt{\pi S} \\\\[0.3ex] &= \sqrt[3]{\pi^2 6 V} \\\\[0.3ex] \end{alignat} }[/math]

## See also

* Arc length - Distance along a curve
* Physics:List of second moments of area - None
* List of trigonometric identities - Equalities that involve trigonometric functions

## References

1. ↑ "Archived copy". Archived from the original on 2012-08-13. https://web.archive.org/web/20120813015606/http://www.austincc.edu/tutor/students/resources/Geometry.pdf. Retrieved 2011-11-29.
2. ↑ "Area Formulas". http://www.math.com/tables/geometry/areas.htm.
3. ↑ "List of Basic Geometry Formulas". 27 May 2018. https://www.andlearning.org/geometry-formulas/.
4. ↑ Treese, Steven A. (2018). History and Measurement of the Base and Derived Units. Cham, Switzerland: Springer Science+Business Media. ISBN 978-3-319-77577-7. OCLC 1036766223.

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