# Rotated Rectangle

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

$$\large{ A_{area} = a \; b }$$     (Area of a Rotated Rectangle)

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

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

Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$  $$\large{ mm^2 }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Distance from Centroid of a Rotated Rectangle formulas

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

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

Symbol English Metric
$$\large{ C }$$ = distance from centroid $$\large{ in^2 }$$  $$\large{ mm^2 }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Elastic Section Modulus of a Rotated Rectangle formula

$$\large{ S_x = \frac{ b\;a\; \left(a^2 \; cos^2 \; \theta \;+\; b^2 sin^2\; \theta \right) }{ 6 \; \left( a \; cos\;\theta \;+\; b\;sin\;\theta\right) } }$$
Symbol English Metric
$$\large{ S }$$ = elastic section modulus $$\large{ in^3 }$$  $$\large{ mm^3 }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Perimeter of a Rotated Rectangle formula

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

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

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

Symbol English Metric
$$\large{ P }$$ = perimeter $$\large{ in }$$  $$\large{ mm }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Polar Moment of Inertia of a Rotated Rectangle formula

$$\large{ J_{z} = \frac{b\;a}{3} \; \left( b^2 \;+\; a^2 \right) }$$
Symbol English Metric
$$\large{ J }$$ = torsional constant $$\large{ in^4 }$$  $$\large{ mm^4 }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Radius of Gyration of a Rotated Rectangle formula

$$\large{ k_{x} = \sqrt{ \frac{ a^2 \;cos^2 \; \left( b^2 \; sin^2 \; \theta \;+\; \theta \right) }{ 2\; \sqrt{3} } } }$$
Symbol English Metric
$$\large{ k }$$ = radius of gyration $$\large{ in }$$  $$\large{ mm }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Second Moment of Area of a Rotated Rectangle formula

$$\large{ k_{x} = \sqrt{ \frac{ a^2 \;cos^2 \; \left( b^2 \; sin^2 \; \theta \;+\; \theta \right) }{ 2\; \sqrt{3} } } }$$
Symbol English Metric
$$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$  $$\large{ mm^4 }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

## Side of a Rotated Rectangle formulas

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

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

Symbol English Metric
$$\large{ P }$$ = perimeter $$\large{ in }$$  $$\large{ mm }$$
$$\large{ a, b }$$ = edge $$\large{ in }$$ $$\large{ mm }$$

Tags: Structural Steel