# Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Trapezoid - Geometric Properties

• Trapezoid is a quadrilateral with only one pair of parallel edges.
• No interior angles are equal.
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of a Trapezoid formula

$$\large{ A_{area} = \frac {h}{2} \left(a \;+\; b\right) }$$

$$\large{ A_{area} = h \left( \frac {a \;+\; b} {2 } \right) }$$

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

### Diagonal of a Trapezoid Formula

$$\large{ d' = \sqrt { b^2 \;+\; d^2 \;-\; 2bd \; cos\;B } }$$

$$\large{ d' = \sqrt { a^2 \;+\; c^2 \;-\; 2ac \; cos\;D } }$$

$$\large{ D' = \sqrt { b^2 \;+\; c^2 \;-\; 2bc \; cos\;A } }$$

$$\large{ D' = \sqrt { d^2 \;+\; a^2 \;-\; 2da \; cos\;C } }$$

### Height of a Trapezoid formula

$$\large{ h = 2 \left( \frac { A_{area} } { a \;+\; b } \right) }$$

$$\large{ h = d \; sin \; C }$$

$$\large{ h = c \; sin \; D }$$

### Perimeter of a Trapezoid formula

$$\large{ P = a \;+\; b \;+\; c \;+\; d }$$

$$\large{ P = \sqrt {h^2 \;+\; g^2} \;+\; \sqrt {h^2 \;+\; \left( b \;-\; a \;-\; g \right)^2 } \;+\; a \;+\; b }$$

### Side of a Trapezoid formula

$$\large{ a = 2 \frac { A_{area} }{h} \;-\; b }$$

$$\large{ b = 2 \frac { A_{area} }{h} \;-\; a }$$

$$\large{ c = P \;-\; a \;-\; b \;-\; d }$$

$$\large{ d = P \;-\; a \;-\; b \;-\; c }$$

### Distance from Centroid of a Trapezoid Formula

$$\large{ C_x = \frac { 2ag \;+\; a^2 \;+\; gb \;+\; ab \;+\; b^2 } { 3 \left( { a \;+\; b } \right) } }$$

$$\large{ C_y = \frac { h } { 3} \left( \frac { 2a \;+\; b } { a \;+\; b } \right) }$$

### Elastic Section Modulus of a Trapezoid formula

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

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

### Plastic Section Modulus of a Trapezoid formula

$$\large{ Z_x = \frac { h^2 \left( 2a^2 \;+\; 14ab \;+\; 2b^2 \right) } { 12 \left( a \;+\; b \right) } }$$

$$\large{ Z_y = \frac { 6abh \;-\; 3a^2h \;-\; 8a \;+\; 8b \;+\; 4g^2h \;-\; 8g } { 24 } }$$

### Polar Moment of Inertia of a Trapezoid formula

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

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

### Radius of Gyration of a Trapezoid formula

$$\large{ k_{x} = \frac { h } { 6 } \sqrt { 2 \;+\; \frac { 4ab} { \left( a \;+\; b \right)^2 } } }$$

$$\large{ k_{y} = \sqrt { \frac {I_y} {A_{area}} } }$$

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

$$\large{ k_{x1} = \frac { 1 } { 6 } \sqrt { \frac { 6h^2 \left( 3a \;+\; b \right) } { a \;+\; b } } }$$

$$\large{ k_{y1} = \sqrt { \frac {I_{y1}} {A_{area}} } }$$

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

### Second Moment of Area of a Trapezoid formula

$$\large{ I_{x} = \frac { h^3 \left( a^2 4ab \;+\; b^2 \right) } { 36 \left( a \;+\; b \right) } }$$

$$\large{ I_{y} = \frac { h \left( 4abg^2 \;+\; 3a^2 bg \;-\; 3ab^2 g \;+\; a^4 \;+\; b^4 \;+\; 2a^3 b \;+\; a^2 g^2 \;+\; a^3 g \;+\; 2ab^3 \;-\; gb^3 \;+\; b^2g^2 \right) } { 36 \left( a \;+\; b \right) } }$$

$$\large{ I_{x1} = \frac { h^3 \left( 3a \;+\; b \right) } { 12 } }$$

$$\large{ I_{y1} = \frac { h \left( a^3 \;+\; 3ag^2 \;+\; 3a^2g \;+\; b^3 \;+\; gb^2 \;+\; ab^2 \;+\; bg^2 \;+\; 2abg \;+\; ba^2 \right) } { 12 } }$$

Where:

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

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

$$\large{ A, B, C, D }$$ = vertex

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

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

$$\large{ h }$$ = height

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

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

$$\large{ P }$$ = perimeter

$$\large{ r }$$ = incircle

$$\large{ R }$$ = outcircle

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

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