# Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A two-dimensional figure that is a quadrilateral that has a pair of parallel opposite sides.
• A trapezoid is a structural shape used in construction.
• See Geometric Properties of Structural Shapes
• No interior angles are equal.
• Angle A & D = 180
• Angle B & C = 180
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of a Trapezoid formula

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

$$\large{ A_{area} = m\;h }$$

Where:

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

$$\large{ h }$$ = height

$$\large{ m }$$ = midline

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

### Diagonal of a Trapezoid Formula

$$\large{ d' = \sqrt{ b^2+c^2-2\;b \sqrt{c^2-h^2} } }$$

$$\large{ D' = \sqrt{ b^2+d^2-2\;b \sqrt{d^2-h^2} } }$$

Where:

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

$$\large{ h }$$ = height

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

### Height of a Trapezoid formula

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

$$\large{ h = \frac { A_{area} } { m } }$$

Where:

$$\large{ h }$$ = height

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

$$\large{ h }$$ = height

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

### Midline of a Trapezoid formula

$$\large{ m = \frac{a + b}{2} }$$

Where:

$$\large{ m }$$ = midline

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

### Perimeter of a Trapezoid formula

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

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

Where:

$$\large{ P }$$ = perimeter

$$\large{ h }$$ = height

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

### Side of a Trapezoid formula

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

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

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

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

Where:

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

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

$$\large{ h }$$ = height

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

### Distance from Centroid of a Trapezoid Formula

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

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

Where:

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

$$\large{ h }$$ = height

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

### Elastic Section Modulus of a Trapezoid 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

### Plastic Section Modulus of a Trapezoid formula

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

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

Where:

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

$$\large{ h }$$ = height

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

### Polar Moment of Inertia of a Trapezoid 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 Trapezoid formula

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

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

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

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

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

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

Where:

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

$$\large{ h }$$ = height

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

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

### Second Moment of Area of a Trapezoid formula

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

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

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

$$\large{ I_{y1} = \frac { h \; \left( a^3 + 3\;a\;g^2 + 3\;a^2\;g + b^3 + g\;b^2 + a\;b^2 + b\;g^2 + 2\;a\;b\;g + b\;a^2 \right) } { 12 } }$$

Where:

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

$$\large{ h }$$ = height

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

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

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