# Isosceles Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Isosceles trapezoid is a trapezoid with only one pair of parallel edges and having base angles that are the same.
• Circumcircle (R) is a circle that passes through all the vertices of a two-dimensional figure.
• x (A, D) = Acute angle measures less than 90°.
• y (B, C) = Obtuse angle measures more than 90°.
• b = d
• A = D, B = C
• AD & BC are bases
• AB & CD are legs
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of an Isosceles Trapezoid formula

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

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

$$\large{ A_{area} = m\;c \; sin \; x }$$

$$\large{ A_{area} = m\;c \; sin \; y }$$

Where:

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

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

$$\large{ h }$$ = height

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

### Circumcircle of an Isosceles Trapezoid formula

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

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

Where:

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

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

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

### Diagonal of an Isosceles Trapezoid Formula

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

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

Where:

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

$$\large{ h }$$ = height

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

### Distance from Centroid of an Isosceles Trapezoid Formula

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

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

Where:

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

$$\large{ h }$$ = height

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

### Elastic Section Modulus of an Isosceles 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 an Isosceles Trapezoid formula

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

Where:

$$\large{ h }$$ = height

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

### Perimeter of an Isosceles Trapezoid formula

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

Where:

$$\large{ P }$$ = perimeter

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

### Plastic Section Modulus of an Isosceles Trapezoid formula

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

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

Where:

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

$$\large{ h }$$ = height

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

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

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

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

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

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

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

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

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 an Isosceles Trapezoid formula

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

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

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

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

Where:

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

$$\large{ h }$$ = height

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

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

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