Square

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • square 13Square (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.
  • See Geometric Properties of Structural Shapes
  • a ∥ c
  • b ∥ d
  • a = b = c = d
  • ∠A = ∠B = ∠C = ∠D = 360°
  • 4 interior angles are 90°
  • 2 diagonals
  • 4 edges
  • 4 vertex

Area of a Square formula

\( \large{ A_{area} = a^2 }\)

\( \large{ A_{area} = \frac{D'^2}{2} }\)

\( \large{ A_{area} = 4\;r^2 }\)

\( \large{ A_{area} = 2\;R^2 }\)

Where:

\(\large{ A_{area} }\) = area

\(\large{ D' }\) = diagonal

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ a, b, c, d }\) = edge

Circumcircle Radius of a Square formula

\(\large{ R =  \frac{a}{ \sqrt {2} }   }\)

\(\large{ R =  \frac{D'}{2}   }\)

Where:

\(\large{ R }\) = outside radius

\(\large{ D' }\) = diagonal

\(\large{ a, b, c, d }\) = edge

Diagonal of a Square formula

\(\large{ D' = a \; \sqrt {2}  }\)

\(\large{ D' = \sqrt {2 \; A_{area} }  }\)

\(\large{ D' = 2\;R  }\)

\(\large{ D' = 2\;r\; \sqrt{2}  }\)

\(\large{ D' = 2\;R  }\)

\(\large{ D' = \frac{P}{ 2\; \sqrt{2} }  }\)

Where:

\(\large{ D' }\) = diagonal

\(\large{ A_{area} }\) = area

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ P }\) = perimeter

\(\large{ a, b, c, d }\) = edge

Distance from Centroid of a Square formula

\(\large{ C_x =  \frac { a }  { 2 }   }\)

\(\large{ C_y =  \frac { a }  { 2}   }\)

Where:

\(\large{ C }\) = distance from centroid

\(\large{ a }\) = side

Elastic Section Modulus of a Square formula

\(\large{ S =  \frac { a^3 }  { 6  }  }\)

Where:

\(\large{ S }\) = elastic section modulus

\(\large{ a, b, c, d }\) = edge

Inscribed Circle Radius of a Square formula

\(\large{ r = \frac{a}{2}   }\)

\(\large{ r = \frac{P}{8}   }\)

\(\large{ r = \frac{D'}{2\; \sqrt{2} }   }\)

\(\large{ r = \frac{ \sqrt{A_{area} } }{2}   }\)

\(\large{ r = \frac{R}{ \sqrt{2} }   }\)

Where:

\(\large{ r }\) = inside radius

\(\large{ A_{area} }\) = area

\(\large{ D' }\) = diagonal

\(\large{ R }\) = outside radius

\(\large{ P }\) = perimeter

\(\large{ a, b, c, d }\) = edge

Perimeter of a Square formula

\(\large{ P = 4\;a  }\)

\(\large{ P = 4\; \sqrt{A_{area} }  }\)

\(\large{ P = 2\;D'\; \sqrt{2}  }\)

\(\large{ P = 4\;R\; \sqrt{2}  }\)

\(\large{ P = 8\;r  }\)

Where:

\(\large{ P }\) = perimeter

\(\large{ A_{area} }\) = area

\(\large{ D' }\) = diagonal

\(\large{ r }\) = inside radius

\(\large{ R }\) = outside radius

\(\large{ a, b, c, d }\) = edge

Side of a Square formula

\(\large{ a = \sqrt  { A_{area}  }   }\)

Where:

\(\large{ a, b, c, d }\) = edge

\(\large{ A_{area} }\) = area

Plastic Section Modulus of a Square formula

\(\large{ Z =  \frac { a^3 }  { 4  }  }\)

Where:

\(\large{ Z }\) = plastic section modulus

\(\large{ a, b, c, d }\) = edge

Polar Moment of Inertia of a Square formula

\(\large{ J_{z} = \frac {a^4}{6}  }\)

\(\large{ J_{z1} =  \frac {2\;a^4}{3}  }\)

Where:

\(\large{ J }\) = torsional constant

\(\large{ a, b, c, d }\) = edge

Radius of Gyration of a Square formula

\(\large{ k_{x} =    \frac { a }  {  2  \sqrt 3  }    }\)

\(\large{ k_{y} =   \frac { a }  {  2  \sqrt 3  }  }\)

\(\large{ k_{z} =   \frac { a }  {  \sqrt 6  }  }\)

\(\large{ k_{x1} =   \frac { a }  {  \sqrt 3  }  }\)

\(\large{ k_{y1} =   \frac { a }  {   \sqrt 3  }  }\)

\(\large{ k_{z1} =   \sqrt {  \frac {2}{3} \;a    }   }\)

Where:

\(\large{ k }\) = radius of gyration

\(\large{ a, b, c, d }\) = edge

Second Moment of Area of a Square formula

\(\large{ I_{x} =  \frac {a^4}{12}  }\)

\(\large{ I_{y} = \frac {a^4}{12}  }\)

\(\large{ I_{x1} =   \frac {a^4}{3}  }\)

\(\large{ I_{y1} =  \frac {a^4}{3}  }\)

Where:

\(\large{ I }\) = moment of inertia

\(\large{ a, b, c, d }\) = edge

Torsional Constant of a Square formula

\(\large{ J  =  2.25  \;  \left(   \frac {  a  } {  2  } \right) ^4  }\)

Where:

\(\large{ J }\) = torsional constant

\(\large{ a, b, c, d }\) = edge

 

Tags: Equations for Moment of Inertia Equations for Structural Steel Equations for Modulus Equations for Geometric Properties