# Isosceles Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Isosceles Trapezoid - Geometric Properties

• Trapezoid is a quadrilateral with only one pair of parallel edges.
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of an Isosceles Trapezoid formula

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

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

$$\large{ A_{area} = mc \; sin \; x }$$

$$\large{ A_{area} = mc \; sin \; y }$$

### Circumcircle of an Isosceles Trapezoid formula

The radius of a circumcircle (outer) of a square if given side of diagonal $$( R )$$.

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

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

### Diagonal of an Isosceles Trapezoid Formula

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

### Height of an Isosceles Trapezoid formula

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

### Perimeter of an Isosceles Trapezoid formula

$$\large{ P= 2c \;+\; a \;+\; b }$$

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

### Distance from Centroid of an Isosceles Trapezoid Formula

$$\large{ C_x = \frac { b } { 2 } }$$

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

### Elastic Section Modulus of an Isosceles Trapezoid formula

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

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

### Plastic Section Modulus of an Isosceles Trapezoid formula

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

$$\large{ Z_y = \frac { h^2 \left( 11a^2 \;+\; 26ab \;+\; 11b^2 \right) } { 48 \left( a \;+\; b \right) } }$$

### Polar Moment of Inertia of an Isosceles Trapezoid formula

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

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

### Radius of Gyration of an Isosceles Trapezoid formula

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

$$\large{ k_{y} = \frac { 1 } { 12 } \sqrt { 6 \left( a^2 \;+\; b^2 \right) } }$$

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

$$\large{ k_{x1} = \frac { h } { 6 } \sqrt { 6 \frac { 3a \;+\; b} { a \;+\; b } } }$$

$$\large{ k_{y1} = \sqrt { \frac { 3b \;+\; 5a} { 12 \left( b \;+\; a \right) } b } }$$

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

### Second Moment of Area of an Isosceles Trapezoid formula

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

$$\large{ I_{y} = \frac { h \left( a \;+\; b \right) \left( a^2 \;+\; b^2 \right) } { 48 } }$$

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

$$\large{ I_{y1} = \frac { h \left( a \;+\; b \right) \left( a^2 \;+\; 7b^2 \right) } { 48 } }$$

Where:

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

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

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

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

$$\large{ D' }$$ = 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