# Square

Written by Jerry Ratzlaff on . 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

## Formulas that use Area of a Square

 $$\large{ A_{area} = a^2 }$$ $$\large{ A_{area} = \frac{D'^2}{2} }$$ $$\large{ A_{area} = 4 \; r^2 }$$ $$\large{ A_{area} = 2 \; R^2 }$$

### Where:

$$\large{ A_{area} }$$ = area

$$\large{ D' }$$ = diagonal

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Circumcircle Radius of a Square

 $$\large{ R = \frac{a}{ \sqrt {2} } }$$ $$\large{ R = \frac{D'}{2} }$$

### Where:

$$\large{ R }$$ = outside radius

$$\large{ D' }$$ = diagonal

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Diagonal of a Square

 $$\large{ D' = a \; \sqrt {2} }$$ $$\large{ D' = \sqrt {2 \; A_{area} } }$$ $$\large{ D' = 2 \; R }$$ $$\large{ D' = 2 \; r \; \sqrt{2} }$$ $$\large{ D' = 2\;R }$$ $$\large{ D' = \frac{P}{ 2\; \sqrt{2} } }$$

### 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

## Formulas that use Distance from Centroid of a Square

 $$\large{ C_x = \frac { a } { 2 } }$$ $$\large{ C_y = \frac { a } { 2} }$$

### Where:

$$\large{ C }$$ = distance from centroid

$$\large{ a }$$ = side

## Formulas that use Elastic Section Modulus of a Square

 $$\large{ S = \frac { a^3 } { 6 } }$$

### Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Inscribed Circle Radius of a Square

 $$\large{ r = \frac{a}{2} }$$ $$\large{ r = \frac{P}{8} }$$ $$\large{ r = \frac{D'}{2\; \sqrt{2} } }$$ $$\large{ r = \frac{ \sqrt{A_{area} } }{2} }$$ $$\large{ r = \frac{R}{ \sqrt{2} } }$$

### 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

## Formulas that use Perimeter of a Square

 $$\large{ P = 4\;a }$$ $$\large{ P = 4\; \sqrt{A_{area} } }$$ $$\large{ P = 2\;D'\; \sqrt{2} }$$ $$\large{ P = 4\;R\; \sqrt{2} }$$ $$\large{ P = 8\;r }$$

### 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

## Formulas that use Side of a Square

 $$\large{ a = \sqrt { A_{area} } }$$

### Where:

$$\large{ a, b, c, d }$$ = edge

$$\large{ A_{area} }$$ = area

## Formulas that use Plastic Section Modulus of a Square

 $$\large{ Z = \frac { a^3 } { 4 } }$$

### Where:

$$\large{ Z }$$ = plastic section modulus

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Polar Moment of Inertia of a Square

 $$\large{ J_{z} = \frac {a^4}{6} }$$ $$\large{ J_{z1} = \frac {2\;a^4}{3} }$$

### Where:

$$\large{ J }$$ = torsional constant

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Radius of Gyration of a Square

 $$\large{ k_{x} = \frac { a } { 2\; \sqrt 3 } }$$ $$\large{ k_{y} = \frac { a } { 2\; \sqrt 3 } }$$ $$\large{ k_{z} = \frac { a } { \sqrt 6 } }$$ $$\large{ k_{x1} = \frac { a } { \sqrt 3 } }$$ $$\large{ k_{y1} = \frac { a } { \sqrt 3 } }$$ $$\large{ k_{z1} = \sqrt { \frac {2}{3} \;a } }$$

### $$\large{ k }$$ = radius of gyration

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Second Moment of Area of a Square

 $$\large{ I_{x} = \frac {a^4}{12} }$$ $$\large{ I_{y} = \frac {a^4}{12} }$$ $$\large{ I_{x1} = \frac {a^4}{3} }$$ $$\large{ I_{y1} = \frac {a^4}{3} }$$

### Where:

$$\large{ I }$$ = moment of inertia

$$\large{ a, b, c, d }$$ = edge

## Formulas that use Torsional Constant of a Square

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

### Where:

$$\large{ J }$$ = torsional constant

$$\large{ a, b, c, d }$$ = edge