# Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Rectangle (a two-dimensional figure) is a quadrilateral with two pair of parallel edges.
• 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.
• 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 = c
• b = d
• ∠A = ∠B = ∠C = ∠D = 360°
• 4 interior angles are 90°
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of a Rectangle formula

$$\large{ A_{area} = a\;b }$$

Where:

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

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

### Circumcircle Radius of a Rectangle formula

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

$$\large{ R = \frac{ \sqrt{ a^2 \;+\; b^2 } }{ 2 } }$$

Where:

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

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

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

### Diagonal of a Rectangle formula

$$\large{ D' = \sqrt {a^2 + b^2 } }$$

Where:

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

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

### Distance from Centroid of a Rectangle formula

$$\large{ C_x = \frac{ b }{ 2 } }$$

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

Where:

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

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

### Elastic Section Modulus of a Rectangle formula

$$\large{ S_x = \frac{ a^2\; b }{ 6 } }$$

$$\large{ S_y = \frac{ a\;b^2 }{ 6 } }$$

Where:

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

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

### Perimeter of a Rectangle formula

$$\large{ P= 2\;a + 2\;b }$$

$$\large{ P= 2\; \left( a + b \right) }$$

Where:

$$\large{ P }$$ = perimeter

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

### Plastic Section Modulus of a Rectangle formula

$$\large{ Z_x = \frac{ a^2 \;b }{ 4 } }$$

$$\large{ Z_y = \frac{ a\;b^2 }{ 4 } }$$

Where:

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

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

### Polar Moment of Inertia of a Rectangle formula

$$\large{ J_{z} = \frac{a\;b}{12} \; \left( a^2 + b^2 \right) }$$

$$\large{ J_{z1} = \frac{a\;b}{3} \; \left( a^2 + b^2 \right) }$$

Where:

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

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

### Radius of Gyration of a Rectangle formula

$$\large{ k_{x} = \frac{ a }{ 2 \; \sqrt{3} } }$$

$$\large{ k_{y} = \frac{ b }{ 2 \; \sqrt{3} } }$$

$$\large{ k_{z} = \sqrt{ \frac{ a^2 \;+\; b^2 }{ 2\; \sqrt{3} } } }$$

$$\large{ k_{x1} = \frac{ a } { \sqrt 3 } }$$

$$\large{ k_{y1} = \frac{ b } { \sqrt 3 } }$$

$$\large{ k_{z1} = \sqrt{ \frac{ a^2 \;+\; b^2 }{ \sqrt{3} } } }$$

Where:

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

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

### Second Moment of Area of a Rectangle formula

$$\large{ I_{x} = \frac{a^3\; b}{12} }$$

$$\large{ I_{y} = \frac{a\;b^3}{12} }$$

$$\large{ I_{x1} = \frac{a^3 \;b}{3} }$$

$$\large{ I_{y1} = \frac{a\;b^3}{3} }$$

Where:

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

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

### Side of a Rectangle formula

$$\large{ a = \frac{P}{2} \;-\; b }$$

$$\large{ b = \frac{P}{2} \;-\; a }$$

Where:

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

$$\large{ P }$$ = perimeter

### Torsional Constant of a Rectangle formula

$$\large{ J = a^3\; b\; \left[ \frac{1}{3} \;-\; \frac{0.21\;a}{b} \; \left( 1 \;-\; \frac{ a^4 }{ 12\;b^4 } \right) \right] }$$

Where:

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

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