Square

Area of a Square formulas
$A_{area} \;=\; a^2$ $A_{area} \;=\; \dfrac{ D'^2 }{ 2 }$ $A_{area} \;=\; 4 \cdot r^2$ $A_{area} \;=\; 2 \cdot R^2$
Symbol	English	Metric
$A_{area}$ = area	$in^2$	$mm^2$
$D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$
$r$ = inside radius	$in$	$mm$
$R$ = outside radius	$in$	$mm$

Square (a two-dimensional figure) is a quadrilateral with four equal side lengths and 90° interior angles.
Abbreviated as SQ
Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
Diagonal is a line from one vertices to another that is non adjacent.
Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
Polygon (a two-dimensional figure) is a closed plane figure for which all edges are line segments and not necessarly congruent.
Quadrilateral (a two-dimensional figure) is a polygon with four sides.

Circumcircle Radius of a Square formulas
$R \;=\; \dfrac{ a }{ \sqrt {2} }$ $R \;=\; \dfrac{ D' }{ 2 }$
Symbol	English	Metric
$R$ = outside radius	$in$	$mm$
$D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$

Diagonal of a Square formulas
$D' \;=\; a \cdot \sqrt {2}$ $D' \;=\; \sqrt {2 \cdot A_{area} }$ $D' \;=\; 2 \cdot R$ $D' \;=\; 2 \cdot r \cdot \sqrt{2}$ $D' \;=\; 2 \cdot R$ $D' \;=\; \dfrac{ P }{ 2 \cdot \sqrt{2} }$
Symbol	English	Metric
$D'$ = diagonal	$in$	$\large{ mm }$
$A_{area}$ = area	$in^2$	$mm^2$
$a, b, c, d$ = edge	$in$	$mm$
$r$ = inside radius	$in$	$mm$
$R$ = outside radius	$in$	$mm$
$P$ = perimeter	$in$	$mm$

Distance from Centroid of a Square formulas
$C_x \;=\; \dfrac{ a }{ 2 }$ $C_y \;=\; \dfrac{ a }{ 2 }$
Symbol	English	Metric
$C$ = distance from centroid	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$

Elastic Section Modulus of a Square formulas
$S \;=\; \dfrac{ a^3 }{ 6 }$
Symbol	English	Metric
$S$ = elastic section modulus	$in^3$	$mm^3$
$a, b, c, d$ = edge	$in$	$mm$

Inscribed Circle Radius of a Square formulas
$r \;=\; \dfrac{ a}{ 2 }$ $r \;=\; \dfrac{ P}{ 8 }$ $r \;=\; \dfrac{ D }{ 2\cdot \sqrt{2} }$ $r \;=\; \dfrac{ \sqrt{A_{area} } }{ 2 }$ $r \;=\; \dfrac{ R}{ \sqrt{2} }$
Symbol	English	Metric
$r$ = inside radius	$in$	$mm$
$A_{area}$ = area	$in^2$	$mm^2$
$D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$
$R$ = outside radius	$in$	$mm$
$P$ = perimeter	$in$	$mm$

Perimeter of a Square formulas
$P \;=\; 4 \cdot a$ $P \;=\; 4\cdot \sqrt{A_{area} }$ $P \;=\; 2\cdot D'\cdot \sqrt{2}$ $P \;=\; 4\cdot R\cdot \sqrt{2}$ $P \;=\; 8\cdot r$
Symbol	English	Metric
$P$ = perimeter	$in$	$mm$
$A_{area}$ = area	$in^2$	$mm^2$
$D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$
$r$ = inside radius	$in$	$mm$
$R$ = outside radius	$in$	$mm$

Plastic Section Modulus of a Square formulas
$Z \;=\; \dfrac{ a^3 }{ 4 }$
Symbol	English	Metric
$Z$ = plastic section modulus	$in^3$	$mm^3$
$a, b, c, d$ = edge	$in$	$mm$

Polar Moment of Inertia of a Square formulas
$J_{z} \;=\; \dfrac{ a^4}{ 6 }$ $J_{z1} \;=\; \dfrac{ 2\cdot a^4}{ 3 }$
Symbol	English	Metric
$J$ = torsional constant	$in^4$	$mm^4$
$a, b, c, d$ = edge	$in$	$mm$

Radius of Gyration of a Square formulas
$k_{x} \;=\; \dfrac{ a }{ 2\cdot \sqrt{3} }$ $k_{y} \;=\; \dfrac{ a }{ 2\cdot \sqrt{3} }$ $k_{z} \;=\; \dfrac{ a }{ \sqrt{6} }$ $k_{x1} \;=\; \dfrac{ a }{ \sqrt{3} }$ $k_{y1} \;=\; \dfrac{ a }{ \sqrt{3} }$ $k_{z1} \;=\; \sqrt{ \frac {2}{3} \cdot a }$
Symbol	English	Metric
$k$ = radius of gyration	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$

Second Moment of Area of a Square formulas
$I_{x} \;=\; \dfrac{ a^4}{ 12 }$ $I_{y} \;=\; \dfrac{ a^4}{ 12 }$ $I_{x1} \;=\; \dfrac{ a^4}{ 3 }$ $I_{y1} \;=\; \dfrac{ a^4}{ 3 }$
Symbol	English	Metric
$I$ = moment of inertia	$in^4$	$mm^4$
$a, b, c, d$ = edge	$in$	$mm$

Torsional Constant formulas
$J \;=\; 2.25 \cdot \left( \dfrac{ a }{ 2 } \right)^4$
Symbol	English	Metric
$J$ = torsional constant	$in^4$	$mm^4$
$a, b, c, d$ = edge	$in$	$mm$