List of formulas in elementary geometry

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

Two-dimensional shapes

Shape	Area	Perimeter/Circumference	Meanings of symbols
Square	$l^{2}$	$4 l$	$l$ is the length of a side
Rectangle	$l b$	$2 (l + b)$	$l$ is length, $b$ is breadth
Circle	$π r^{2}$	$2 π r$ or $π d$	where $r$ is the radius and $d$ is the diameter
Ellipse	$π a b$		where $a$ is the semimajor axis and $b$ is the semiminor axis
Triangle	$\frac{b h}{2}$	$a + b + c$	$b$ is base; $h$ is height; $a, b, c$ are sides
Parallelogram	$b h$	$2 (a + b)$	$b$ is base, $h$ is height, $a$ is side
Trapezoid	$\frac{a + b}{2} h$		$a$ and $b$ 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]^{: 405–406}

Cone – $\frac{1}{3} π r^{2} h$ , where $r$ is the base's radius and $h$ is the cone's height;
Cube – $a^{3}$ , where $a$ is the side's length;
Cuboid – $a b c$ , where $a$ , $b$ , and $c$ are the sides' length;
Cylinder – $π r^{2} h$ , where $r$ is the base's radius and $h$ is the cylinder's height;
Ellipsoid – $\frac{4}{3} π a b c$ , where $a$ , $b$ , and $c$ are the semi-major and semi-minor axes' length;
Sphere – $\frac{4}{3} π r^{3}$ , where $r$ is the radius;
Parallelepiped – $a b c \sqrt{K}$ , where $a$ , $b$ , and $c$ are the sides' length, $K = 1 + 2 \cos (α) \cos (β) \cos (γ) - \cos^{2} (α) - \cos^{2} (β) - \cos^{2} (γ)$ , and $α$ , $β$ , and $γ$ are angles between the two sides;
Prism – $B h$ , where $B$ is the base's area and $h$ is the prism's height;
Pyramid – $\frac{1}{3} B h$ , where $B$ is the base's area and $h$ is the pyramid's height;
Tetrahedron – $\frac{\sqrt{2}}{12} a^{3}$ , where $a$ 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

$r$ is the radius,
$C = 2 π r$ is the circumference (the length of any one of its great circles),
$S$ is the surface area,
$V$ is the volume.

Surface area:

$\begin{matrix} S & = 4 π r^{2} \\ = \frac{1}{π} C^{2} \\ = \sqrt[3]{π (6 V)^{2}} \end{matrix}$

Volume:

$\begin{matrix} V & = \frac{4}{3} π r^{3} \\ = \frac{1}{6 π^{2}} C^{3} \\ = \frac{1}{6 \sqrt{π}} S^{3 / 2} \end{matrix}$

Radius:

$\begin{matrix} r & = \frac{1}{2 π} C \\ = \sqrt{\frac{1}{4 π} S} \\ = \sqrt[3]{\frac{3}{4 π} V} \end{matrix}$

Circumference:

$\begin{matrix} C & = 2 π r \\ = \sqrt{π S} \\ = \sqrt[3]{π^{2} 6 V} \end{matrix}$

References

↑ "Archived copy" (PDF). Archived from the original (PDF) on 2012-08-13. Retrieved 2011-11-29.{{cite web}}: CS1 maint: archived copy as title (link)
↑ "Area Formulas".
↑ "List of Basic Geometry Formulas". 27 May 2018.
↑ 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. LCCN 2018940415. OCLC 1036766223.

[1] "Archived copy" (PDF). Archived from the original (PDF) on 2012-08-13. Retrieved 2011-11-29.{{cite web}}: CS1 maint: archived copy as title (link)

[2] "Area Formulas".

[3] "List of Basic Geometry Formulas". 27 May 2018.

[:2-4] 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. LCCN 2018940415. OCLC 1036766223.

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[2]

[3]

[4]

Two-dimensional shapes

Three-dimensional shapes

Sphere

See also

References