Square

on 29 January 2016. Posted in Plane Geometry

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.
a ∥ c
b ∥ d
a = b = c = d
∠A = ∠B = ∠C = ∠D = 360°
4 interior angles are 90°
2 diagonals
4 edges
4 vertex

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

Diagonal of a Square formulas
\( D' \;=\; a \; \sqrt {2} \) \( D' \;=\; \sqrt {2 \; A_{area} } \) \( D' \;=\; 2 \; R \) \( D' \;=\; 2 \; r \; \sqrt{2} \) \( D' \;=\; 2\;R \) \( D' \;=\; P\;/\; 2\; \sqrt{2} \)
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
\( C_x \;=\; a \;/\; 2 \) \( C_y \;=\; a \;/\; 2 \)
Symbol	English	Metric
\( C \) = distance from centroid	\( in \)	\( mm \)
\( a, b, c, d \) = edge	\( in \)	\( mm \)

Elastic Section Modulus of a Square formulas
\( S \;=\; a^3 \;/\; 6 \)
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
\( r \;=\; a\;/\;2 \) \( r \;=\; P\;/\;8 \) \( r \;=\; D\;/\;2\; \sqrt{2} \) \( r \;=\; \sqrt{A_{area} } \;/\;2 \) \( r \;=\; R\;/\; \sqrt{2} \)
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
\( P = 4\;a \) \( P = 4\; \sqrt{A_{area} } \) \( P = 2\;D'\; \sqrt{2} \) \( P = 4\;R\; \sqrt{2} \) \( P = 8\;r \)
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
\( Z = a^3 \;/\; 4 \)
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
\( J_{z} = a^4\;/\;6 \) \( J_{z1} = 2\;a^4\;/\;3 \)
Symbol	English	Metric
\( J \) = torsional constant	\( in^4 \)	\( mm^4 \)
\( a, b, c, d \) = edge	\( in \)	\( mm \)

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

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

Torsional Constant formulas
\( J = 2.25 \; ( a \;/\; 2 )^4 \)
Symbol	English	Metric
\( J \) = torsional constant	\( in^4 \)	\( mm^4 \)
\( a, b, c, d \) = edge	\( in \)	\( mm \)

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