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Parallelograms and Quadrilaterals

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What is a Parallelogram?

A Parallelogram is defined as a four-sided plane object having opposite sides that are parallel and equal. As plane geometry figures they are flat objects on a level surface with length or height and width or base dimensions. All Parallelograms are classified as Quadrilateral.

Finding the Area of Parallelograms

Finding Perimeters of Parallelograms

Quadrilaterals

Excellent types of Parallelogram shapes are Rectangle, Square and Rhombus.

What are Parallelograms?Parallelogram AreaPerimeter of a ParallelogramWhat is a Quadrilateral?

The formula for the area of any Parallelogram is:

Area = Width x Length or Base x Height

Area of Square

Length or Height

Width or Base

Length or Height

Width or Base

Area of Rectangle

The Rectangle is the ideal definition of a Parallelogram with its parallel opposite sides of equal length. Opposite sides are congruent and all corners form congruent 90 degree angles.

Square objects differ from all other Parallelograms by having all sides of equal length and each corner a 90 degree angle. All sides and all corners are congruent.

Area of Rhombus

Width or Base

Length or Height

The Rhombus is a special type of Parallelogram where opposite sides are parallel, all four sides are equal length and opposite angles are equal. Opposite sides and opposite corners are congruent.

The sum of angles of a Parallelogram equals 360 degrees. For the Rectangle or Square the sum total of four corners, each a 90 degree angle, give us 360 degrees.

A rhombus is different, opposite corners are equal. As a result, two opposite corners of a Rhombus will each have angles less than 90 degrees and two opposite corners will each have angles greater than 90 degrees. The more a Rhombus is leaned the larger two opposite angles become while the other two opposite angles become a like amount smaller. The sum of two opposing angles is always less  than 180 degrees and greater than 0 degrees.

By knowing the corner to corner measurement of any side, as a straight line, and the degrees of angle at any corner we can attain the total area of a Rhombus. To explain how, suppose the red Rhombus has two opposing angles each 75° , the other opposing angles each are 105°. Also suppose the measurement of any line side from corner to corner is 5.  The area can be determined by Trigonometric Sin:

Sin 75° = .965926

Sin 105° = .965926

Rhombus Area =  5 x 5 x .965926 = 24.14815

The same Trigonometric method can be applied to obtain total area of a Square object, but is unnecessary. Each corner of a Square is 90° and the Sin 90° is 1. To find the area of a Rectangle this method cannot be used. By definition a Rectangle’s Length or Height measurement differs from its Width or Base.

How to find the Area of a Rhombus.How to find the Area of a Parallelogram.Finding the Area of a Rectangle.

Perimeter is a total border measurement of a plane geometric figure. To find the perimeter of a Parallelogram all boundary sides are summed. If we suppose the measurement of each side of  Rhombus or Square is 5 the Perimeter for the Rhombus or Square figure is 20:

Perimeter is a border measurement of an area.

Perimeter is a border measurement:

Area is inside a Perimeter.

Area measures extent inside a shape’s borders:

Finding Perimeter of a Parallelogram.

Perimeter = S1 + S2 + S3 + S4

Finding Perimeter of a Rectangle.

A Quadrilateral is a plane figure with four sides and four angles. The less popular synonyms Tetragon and Quadrangle are sometimes used to describe a Quadrilateral shaped figure.

Ordinary Trapezoid does not have symmetry of Isosceles Trapezoid.

While all Parallelograms are Quadrilateral, not all Quadrilaterals are a Parallelogram. Types of Quadrilateral Shape that are not a Parallelogram is Trapezoid and Trapezium.

Trapezoid:

Isosceles Trapezoid:

Isosceles Trapezoid is a Quadrilateral.

To discover the Area of the Isosceles Trapezoid we must first understand how to measure for its width or base. By placing both triangles onto one side creates a Rectangle. The width of the Rectangle always equals ½ the sum of line sides b and b2, giving us width or base of an Isosceles Trapezoid.

The diagonals (line sides a and c), of an Isosceles Trapezoid are always congruent. This creates two triangles that are, except for orientation, identical. It also means the Trapezoid sides labeled b and b2 are parallel.

Isosceles Trapezoid highlighting left and right triangles in red:

Geometric symmetry of Isosceles Trapezoid.

An ordinary Trapezoid does not have the symmetry of the Isosceles Trapezoid.

A Trapezium is a Quadrilateral without parallel sides.

Trapezium:

Trapezium is any Quadrilateral shape lacking parallel sides.Area of an Isosceles Trapezoid by Anayltical Geometry.

Width or Base of Isosceles Trapezoid = ½ (b + b2)

Area = Width x Length or Base x Height = ½ (b + b2) h

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