# Isosceles Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Isosceles trapezoid (a two-dimensional figure) is a trapezoid with only one pair of parallel edges and having base angles that are the same.
• Acute angle measures less than 90°.
• Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
• Diagonal is a line from one vertices to another that is non adjacent.
• Obtuse angle measures more than 90°.
• a & c are bases
• b & d are legs
• a ∥ c
• a ≠ c
• b = d
• ∠A & ∠D < 90°
• ∠B & ∠C > 90°
• ∠A = ∠D
• ∠B = ∠C
• ∠A + ∠B = 180°
• ∠C + ∠D = 180°
• ∠A + ∠C = 180°
• ∠B + ∠D = 180°
• 2 diagonals
• 4 edges
• 4 vertexs

## formulas that use Angle of a Isosceles Trapezoid

 $$\large{ x = arccos \; \frac{\left(\frac{a \;-\; c}{2}\right)^2 \;+\; b^2 \;-\; h^2}{2\;\left(\frac{a \;-\; c}{2}\right)\;b} }$$ $$\large{ y = 180° - x }$$

### Where:

$$\large{ x }$$ = acute angle

$$\large{ y }$$ = obtuse angle

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

$$\large{ h }$$ = height

## formulas that use Area of an Isosceles Trapezoid

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

### Where:

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

$$\large{ m }$$ = central median

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

$$\large{ h }$$ = height

## formulas that use Circumcircle of an Isosceles Trapezoid

 $$\large{ R = \frac{ b\;D'\;c }{ 4 \;\sqrt { s \; \left( s \;-\; b \right) \; \left( s \;-\; D' \right) \left( s \;-\; c \right) } } }$$          $$\large{ s = \frac {b \;+\; D' \;+\; c} {2} }$$ $$\large{ R = \frac { b\; D' \;a } { 4 \;\sqrt { s \; \left( s \;-\; b \right) \; \left( s \;-\; D' \right) \left( s \;-\; a \right) } } }$$          $$\large{ s = \frac {b \;+\; D' \;+\; a} {2} }$$ $$\large{ R = b \; \sqrt{ \frac{a\;c \;+\; b^2}{4\;b^2 \;-\; \left( a \;-\; c \right)^2} } }$$

### Where:

$$\large{ R }$$ = outside radius

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

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

## formulas that use Diagonal of an Isosceles Trapezoid

 $$\large{ D' = \sqrt { b^2 \;+\; c\;a } }$$

### Where:

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

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

$$\large{ h }$$ = height

## formulas that use Distance from Centroid of an Isosceles Trapezoid

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

### Where:

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

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

$$\large{ h }$$ = height

## formulas that use Elastic Section Modulus of an Isosceles Trapezoid

 $$\large{ S_x = \frac { I_x } { C_y } }$$ $$\large{ S_y = \frac { I_y } { C_x } }$$

### Where:

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

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

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

## formulas that use Height of an Isosceles Trapezoid

 $$\large{ h = \frac{1}{2} \; \sqrt { 4\;b^3 \;-\; {c \;+\; a} } }$$

### Where:

$$\large{ h }$$ = height

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

## formulas that use Perimeter of an Isosceles Trapezoid

 $$\large{ P = 2\;b + c + a }$$ $$\large{ P = 2 \; \sqrt{ h^2 + \frac{\left(b \;-\; a\right)^2}{4} } + a + b }$$

### Where:

$$\large{ P }$$ = perimeter

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

$$\large{ h }$$ = height

## formulas that use Plastic Section Modulus of an Isosceles Trapezoid

 $$\large{ Z_x = \frac{ h \; \left( 2\;c^2 \;-\; c\;a \;+\; 2\;a^2 \right) }{ 12 } }$$ $$\large{ Z_y = \frac{ h^2 \; \left( 11\;c^2 \;+\; 26\;c\;a \;+\; 11\;a^2 \right) }{ 48 \; \left( c \;+\; a \right) } }$$

### Where:

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

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

$$\large{ h }$$ = height

## formulas that use Polar Moment of Inertia of an Isosceles Trapezoid

 $$\large{ J_{z} = I_x + I_y }$$ $$\large{ J_{z1} = I_{x1} + I_{y1} }$$

### Where:

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

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

## formulas that use Radius of Gyration of an Isosceles Trapezoid

 $$\large{ k_{x} = \frac{ h }{ 6 } \; \sqrt{ 2 + \frac{ 4\;c\;a}{ \left( c \;+\; a \right)^2 } } }$$ $$\large{ k_{y} = \frac{ 1 }{ 12 } \; \sqrt{ 6 \left( c^2 + a^2 \right) } }$$ $$\large{ k_{z} = \sqrt{ k_{x}{^2} + k_{y}{^2} } }$$ $$\large{ k_{x1} = \frac{ h }{ 6 } \; \sqrt{ 6 \; \frac{ 3\;c \;+\; a}{ c \;+\; a } } }$$ $$\large{ k_{y1} = \sqrt{ \frac{ 3\;a \;+\; 5\;c}{ 12\; \left( a \;+\; c \right) } \;a } }$$ $$\large{ k_{z1} = \sqrt{ k_{x1}{^2} + k_{y1}{^2} } }$$

### Where:

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

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

$$\large{ h }$$ = height

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

## formulas that use Second Moment of Area of an Isosceles Trapezoid

 $$\large{ I_{x} = \frac{ h^3 \; \left( c^2 \;+\; 4\;c\;a \;+\; a^2 \right) }{ 36 \; \left( c \;+\; a \right) } }$$ $$\large{ I_{y} = \frac{ h \; \left( c \;+\; a \right) \left( c^2 \;+\; a^2 \right) }{ 48 } }$$ $$\large{ I_{x1} = \frac{ h^3 \; \left( 3\;c \;+\; a \right) }{ 12 } }$$ $$\large{ I_{y1} = \frac{ h \; \left( c \;+\; a \right) \left( c^2 \;+\; 7\;a^2 \right) }{ 48 } }$$

### Where:

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

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

$$\large{ h }$$ = height

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

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