# 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.
• 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