# Parallelogram

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Parallelogram (a two-dimensional figure) is a quadrilateral with two pairs of parallel opposite sides.
• Acute angle measures less than 90°.
• Diagonal is a line from one vertices to another that is non adjacent.
• Obtuse angle measures more than 90°.
• Opposite sides are congurent and parallel.
• Polygon (a two-dimensional figure) is a closed plane figure for which all edges are line segments and not necessarly congruent.
• Quadrilateral (a two-dimensional figure) is a polygon with four sides.
• a ∥ c
• b ∥ d
• ∠A & ∠C < 90°
• ∠B & ∠D > 90°
• ∠A + ∠B = 180°
• ∠C + ∠D = 180°
• 2 diagonals
• 4 edges
• 4 vertexs

### Angle of a Parallelogram formula

$$\large{ cos \; x = \frac {a^2 \;+\; b^2 \;-\; d'^2 }{2\;a\;b} }$$

$$\large{ cos \; y = \frac {a^2 \;+\; b^2 \;-\; D'^2 }{2\;a\;b} }$$

$$\large{ sin \; x = sin \; y \; \frac {A }{a\;b} }$$

Where:

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

$$\large{ y }$$ = obtuce angles

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

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

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

### Area of a Parallelogram formula

$$\large{ A_{area} = a\;h_a }$$

$$\large{ A_{area} = b\;h_b }$$

$$\large{ A_{area} = a\;b \; sin \;x }$$

$$\large{ A_{area} = a\;b \; sin \;y }$$

Where:

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

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

$$\large{ h_a, h_b }$$ = height

### Diagonal of a Parallelogram formula

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

$$\large{ d' = \sqrt{ a^2 \;+\; b^2 \;+\; 2\;a\;b \; cos \; y } }$$

$$\large{ D' = \sqrt{ a^2 \;+\; b^2 \;-\; 2\;a\;b \; cos \; y } }$$

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

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

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

Where:

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

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

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

$$\large{ y }$$ = obtuce angles

### Edge of a Parallelogram formula

$$\large{ a = \frac {P}{2} - b }$$

$$\large{ b = \frac {P}{2} - a }$$

$$\large{ b = \frac {A}{h} }$$

$$\large{ a = \frac {h_b}{sin\; x} }$$

$$\large{ a = \frac {h_b}{sin\; y} }$$

$$\large{ b = \frac {h_a}{sin\; x} }$$

$$\large{ b = \frac {h_a}{sin\; y} }$$

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

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

Where:

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

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

$$\large{ h_a, h_b }$$ = height

$$\large{ P }$$ = perimeter

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

$$\large{ y }$$ = obtuce angles

### Height of a Parallelogram formula

$$\large{ h_a = \frac {A}{b} }$$

$$\large{ h_a = b \; sin \; x }$$

$$\large{ h_a = b \; sin \; y }$$

$$\large{ h_b = a \; sin \; x }$$

$$\large{ h_b = a \; sin \; y }$$

Where:

$$\large{ h_a, h_b }$$ = height

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

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

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

$$\large{ y }$$ = obtuce angles

### Perimeter of a Parallelogram formula

$$\large{ P = 2 \left( a+b \right) }$$

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

$$\large{ P = 2\;a + \sqrt{ D'^2 + d'^2 - 4\;a^2 } }$$

$$\large{ P = 2\;b + \sqrt{ D'^2 + d'^2 - 4\;b^2 } }$$

Where:

$$\large{ P }$$ = perimeter

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

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