# Rotated Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Rectangle is a quadrilateral with two pair of parallel lines.
• A rotated 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

### Area of a Rotated Rectangle formula

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

Where:

$$\large{ A }$$ = area

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

### Distance from Centroid of a Rotated Rectangle formula

$$\large{ C_x = \frac { b \; cos \; \theta \;+\; a \; sin \; \theta } { 2 } }$$

$$\large{ C_y = \frac { a \; cos \; \theta \;+\; b \; sin \; \theta } { 2 } }$$

Where:

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

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

### Elastic Section Modulus of a Rotated Rectangle formula

$$\large{ S_x = \frac{ b\;a\; \left(a^2 \; cos^2 \; \theta \;+\; b^2 sin^2\; \theta \right) }{ 6 \; \left( a \; cos\;\theta \;+\; b\;sin\;\theta\right) } }$$

Where:

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

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

### Perimeter of a Rotated Rectangle formula

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

Where:

$$\large{ P }$$ = perimeter

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

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

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

Where:

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

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

### Radius of Gyration of a Rotated Rectangle formula

$$\large{ k_{x} = \sqrt{ \frac{ a^2 \;cos^2 \; \left( b^2 \; sin^2 \; \theta \;+\; \theta \right) }{ 2\; \sqrt{3} } } }$$

Where:

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

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

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

$$\large{ I_{x} = \frac{ba}{12} \; \left( a^2 \; cos^2 \; \theta + b^2 \; sin^2 \; \theta \right) }$$

Where:

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

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

### Side of a Rotated Rectangle formula

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

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

Where:

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

$$\large{ P }$$ = perimeter