# 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.
• See Geometric Properties of Structural Shapes
• Interior angles are 90°
• Exterior angles are 90°
• Angle $$\;A = B = C = D$$
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of a Hollow Rectangle formula

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

Where:

$$\large{ A }$$ = area

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

### Perimeter of a Hollow Rectangle formula

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

$$\large{ P= 2\; \left( a_1 + b_2 \right) }$$   ( Inside )

Where:

$$\large{ P }$$ = perimeter

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

### Side of a Hollow Rectangle formula

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

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

Where:

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

$$\large{ P }$$ = perimeter

### Distance from Centroid of a Hollow Rectangle formula

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

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

Where:

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

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

### Elastic Section Modulus of a Hollow Rectangle formula

$$\large{ S_x = \frac { I_x } { C_y } }$$

$$\large{ S_y = \frac { I_y } { C_x } }$$

Where:

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

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

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

### Polar Moment of Inertia of a Hollow Rectangle formula

$$\large{ J_{z} = I_x + I_y }$$

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

Where:

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

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

### Radius of Gyration of a Hollow Rectangle formula

$$\large{ k_{x} = \sqrt{ \frac { b\; a^3 - b_1 \;a_{1}{^3} } { 12 \; \left( b\;a - b_1\; a_1 \right) } } }$$

$$\large{ k_{y} = \sqrt{ \frac { b^3 \;a - b_{1}{^3} \; a_1 } { 12\; \left( b\;a - b_1\; a_1 \right) } } }$$

$$\large{ k_{z} = \sqrt { k_{x}{^2} + k_{y}{^2} } }$$

$$\large{ k_{x1} = \sqrt{ \frac { I_{x1} } { A } } }$$

$$\large{ k_{y1} = \sqrt{ \frac { I_{y1} } { A } } }$$

$$\large{ k_{z1} = \sqrt { k_{x1}{^2} + k_{y1}{^2} } }$$

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

### Second Moment of Area of a Hollow Rectangle formula

$$\large{ I_{x} = \frac { b \;a^3 - b_1\; a_{1}{^3} } {12} }$$

$$\large{ I_{y} = \frac { b^3 \;a - b_{1}{^3}\; a_1 } {12} }$$

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

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

Where:

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

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