Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Trapezoid (a two-dimensional figure) is a quadrilateral that has a pair of parallel opposite sides.
• Acute angle measures less than 90°.
• Diagonal is a line from one vertices to another that is non adjacent.
• No interior angles are equal.
• Obtuse angle measures more than 90°.
• Quadrilateral (a two-dimensional figure) is a polygon with four sides.
• See Geometric Properties of Structural Shapes
• a & c are bases
• b & d are legs
• a ∥ c
• a ≠ c
• ∠A + ∠B = 180°
• ∠C + ∠D = 180°
• 2 diagonals
• 4 edges
• 4 vertexs

Area of a Trapezoid formula

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

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

Where:

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

$$\large{ h }$$ = height

$$\large{ m }$$ = midline

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

Diagonal of a Trapezoid Formula

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

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

Where:

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

$$\large{ h }$$ = height

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

Distance from Centroid of a Trapezoid Formula

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

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

Where:

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

$$\large{ h }$$ = height

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

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

Height of a Trapezoid formula

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

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

Where:

$$\large{ h }$$ = height

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

$$\large{ h }$$ = height

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

Midline of a Trapezoid formula

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

Where:

$$\large{ m }$$ = midline

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

Perimeter of a Trapezoid formula

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

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

Where:

$$\large{ P }$$ = perimeter

$$\large{ h }$$ = height

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

Side of a Trapezoid formula

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

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

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

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

Where:

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

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

$$\large{ h }$$ = height

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

Plastic Section Modulus of a Trapezoid formula

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

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

Where:

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

$$\large{ h }$$ = height

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

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

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

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

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

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

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

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

Where:

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

$$\large{ h }$$ = height

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

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

Second Moment of Area of a Trapezoid formula

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

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

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

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

Where:

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

$$\large{ h }$$ = height

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

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

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