Parallelogram
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