Parallelogram

on 29 January 2016. 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 formulas
\(\large{ cos \; x = \frac {a^2 \;+\; b^2 \;-\; d'^2 }{2\;a\;b} }\) \(\large{ cos \; y = \frac {a^2 \;+\; b^2 \;-\; D'^2 }{2\;a\;b} }\) \(\large{ sin \; x = sin \; y \; \frac {A }{a\;b} }\)
Symbol	English	Metric
\(\large{ x }\) = acute angles	\(\large{ deg}\)	\(\large{ rad}\)
\(\large{ y }\) = obtuce angles	\(\large{ deg}\)	\(\large{ rad}\)
\(\large{ d', D' }\) = diagonal	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ A, B, C, D }\) = vertex	\(\large{ in}\)	\(\large{ mm }\)

Area of a Parallelogram formulas
\(\large{ A_{area} = a\;h_a }\) \(\large{ A_{area} = b\;h_b }\) \(\large{ A_{area} = a\;b \; sin \;x }\) \(\large{ A_{area} = a\;b \; sin \;y }\)
Symbol	English	Metric
\(\large{ A_{area} }\) = area	\(\large{ in^2}\)	\(\large{ mm^2 }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ h_a, h_b }\) = height	\(\large{ in}\)	\(\large{ mm }\)

Diagonal of a Parallelogram formulas
\(\large{ d' = \sqrt{ a^2 \;+\; b^2 \;-\; 2\;a\;b \; cos \; x } }\) \(\large{ d' = \sqrt{ a^2 \;+\; b^2 \;+\; 2\;a\;b \; cos \; y } }\) \(\large{ D' = \sqrt{ a^2 \;+\; b^2 \;-\; 2\;a\;b \; cos \; y } }\) \(\large{ D' = \sqrt{ a^2 \;+\; b^2 \;+\; 2\;a\;b \; cos \; x } }\) \(\large{ d' = \sqrt{ 2\;a^2 \;+\; 2\;b^2 \;-\; D'^2 } }\) \(\large{ D' = \sqrt{ 2\;a^2 \;+\; 2\;b^2 \;-\; d'^2 } }\)
Symbol	English	Metric
\(\large{ d', D' }\) = diagonal	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ x }\) = acute angles	\(\large{ deg}\)	\(\large{ rad}\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ y }\) = obtuce angles	\(\large{ deg}\)	\(\large{ rad}\)

Edge of a Parallelogram formulas
\(\large{ a = \frac {P}{2} - b }\) \(\large{ b = \frac {P}{2} - a }\) \(\large{ b = \frac {A}{h} }\) \(\large{ a = \frac {h_b}{sin\; x} }\) \(\large{ a = \frac {h_b}{sin\; y} }\) \(\large{ b = \frac {h_a}{sin\; x} }\) \(\large{ b = \frac {h_a}{sin\; y} }\) \(\large{ a = \sqrt{ \frac {D'^2 \;+\; d'^2 \;-\; 2\;b^2 }{2} } }\) \(\large{ b = \sqrt{ \frac {D'^2 \;+\; d'^2 \;-\; 2\;a^2 }{2} } }\)
Symbol	English	Metric
\(\large{ a, b, c, d }\) = edge	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ x }\) = acute angles	\(\large{ deg}\)	\(\large{ rad}\)
\(\large{ d', D' }\) = diagonal	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ h_a, h_b }\) = height	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ y }\) = obtuce angles	\(\large{ deg}\)	\(\large{ rad}\)
\(\large{ P }\) = perimeter	\(\large{ in}\)	\(\large{ mm }\)

Height of a Parallelogram formulas
\(\large{ h_a = \frac {A}{b} }\) \(\large{ h_a = b \; sin \; x }\) \(\large{ h_a = b \; sin \; y }\) \(\large{ h_b = a \; sin \; x }\) \(\large{ h_b = a \; sin \; y }\)
Symbol	English	Metric
\(\large{ h_a, h_b }\) = height	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ x }\) = acute angles	\(\large{ deg}\)	\(\large{ rad}\)
\(\large{ A_{area} }\) = area	\(\large{ in^2}\)	\(\large{ mm^2 }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ y }\) = obtuce angles	\(\large{ deg}\)	\(\large{ rad}\)

Perimeter of a Parallelogram formulas
\(\large{ P = 2 \left( a+b \right) }\) \(\large{ P = 2\;a + 2\;b }\) \(\large{ P = 2\;a + \sqrt{ D'^2 + d'^2 - 4\;a^2 } }\) \(\large{ P = 2\;b + \sqrt{ D'^2 + d'^2 - 4\;b^2 } }\)
Symbol	English	Metric
\(\large{ P }\) = perimeter	\(\large{ in}\)	\(\large{ mm}\)
\(\large{ d', D' }\) = diagonal	\(\large{ in}\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in}\)	\(\large{ mm }\)

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Angle of a Parallelogram formulas