# Hollow Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A two-dimensional figure that is a quadrilateral with two pair of parallel edges.
• A hollow rectangle is a structural shape used in construction.
• Interior angles are 90°
• Exterior angles are 90°
• Angle $$\;A = B = C = D$$
• 2 diagonals
• 4 edges
• 4 vertexs

## Formulas that use Area of a Hollow Rectangle

 $$\large{ A = b\;a - b_1\;a_1 }$$

### Where:

$$\large{ A }$$ = area

$$\large{ a, b, a_1, b_1 }$$ = side

## Formulas that use Distance from Centroid of a Hollow Rectangle

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

### Where:

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

$$\large{ a, b, a_1, b_1 }$$ = side

## Formulas that use Elastic Section Modulus of a Hollow Rectangle

 $$\large{ S_x = \frac{ I_x }{ C_y } }$$ $$\large{ S_y = \frac{ I_y }{ C_x } }$$

### Where:

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

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

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

## Formulas that use Perimeter of a Hollow Rectangle

 $$\large{ P_o = 2\; \left( a + b \right) }$$ (outside) $$\large{ P_i = 2\; \left( a_1 + b_2 \right) }$$ (inside)

### Where:

$$\large{ P }$$ = perimeter

$$\large{ a, b, a_1, b_1 }$$ = side

## Formulas that use Polar Moment of Inertia of a Hollow Rectangle

 $$\large{ J_{z} = I_x + I_y }$$ $$\large{ J_{z1} = I_{x1} + I_{y1} }$$

### Where:

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

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

## Formulas that use Radius of Gyration of a Hollow Rectangle

 $$\large{ k_{x} = \sqrt{ \frac{ b\;a^3 \;-\; b_1\; a_{1}{^3} }{ 12 \; \left( b\;a \;-\; b_1\; a_1 \right) } } }$$ $$\large{ k_{y} = \sqrt{ \frac{ b^3 \;a \;-\; b_{1}{^3} \; a_1 }{ 12\; \left( b\;a \;-\; b_1\; a_1 \right) } } }$$ $$\large{ k_{z} = \sqrt{ k_{x}{^2} + k_{y}{^2} } }$$ $$\large{ k_{x1} = \sqrt{ \frac{ I_{x1} }{ A } } }$$ $$\large{ k_{y1} = \sqrt{ \frac{ I_{y1} }{ A } } }$$ $$\large{ k_{z1} = \sqrt{ k_{x1}{^2} + k_{y1}{^2} } }$$

### Where:

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

$$\large{ A }$$ = area

$$\large{ a, b, a_1, b_1 }$$ = side

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

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

## Formulas that use Second Moment of Area of a Hollow Rectangle

 $$\large{ I_{x} = \frac{ b\;a^3 \;-\; b_1\; a_{1}{^3} }{12} }$$ $$\large{ I_{y} = \frac{ b^3 \;a \;-\; b_{1}{^3}\; a_1 }{12} }$$ $$\large{ I_{x1} = \frac{ b\;a^3 }{3} - \frac { b_1 \; a_1 \; \left( a_{1}{^2} \;+\; 3\;a^2 \right) }{12} }$$ $$\large{ I_{y1} = \frac{ b^3 \;a }{3} - \frac { b_1 \; a_1 \; \left( b_{1}{^2} \;+\; 3\;b^2 \right) }{12} }$$

### Where:

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

$$\large{ a, b, a_1, b_1 }$$ = side

## Formulas that use Side of a Hollow Rectangle

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

### Where:

$$\large{ a, b, a_1, b_1 }$$ = side

$$\large{ P }$$ = perimeter