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40 Quadrilaterals

Quadrilaterals are polygons with four sides.

Vocabulary: Click on the words below for their definitions

The Types of Quadrilaterals with Equal Sides

Square

  • Four equal sides or edges
  • Four equal angles

 

If the shape has four equal sides and four equal angles, it’s a square.

The direction of the shape below has several names:

Photograph of a princess-cut diamond on point with green plant background
Princess-cut diamond on point. Photograph by Seth Lemmons. CC-BY-SA 2.0.
  • Square on point
  • Square on tip
  • Diamond shape
  • Skewed square

 

Watch how a skewed square is drawn inside a circle. The square is inscribed in the circle.

 

Skewed Square Inscribed in a Circle

Rhombus

  • Four equal sides or edges
  • Two pair of equal angles (opposite angles are equivalent)
line drawing of a rhombus indicating that all sides are equal
Rhombus has equal sides or edges
red line rhombus indicating opposite angles are equal
Rhombus has opposite angles that are equal
Rhombus polygon with equal sides and two pair of equal angles

 

 

 

 

 

 

 

 

Rombus tangential animation
An animation of square into a rhombus. The radius of the inscribed circle is calculated as the shape changes. The side angle is also calculated through the movement. Source: J Hokkanen, CC-BY-SA, 3.0.

 

The plural for rhombus is rhombi because it is originally a Greek word.

 

Your Turn

 

Types of Quadrilaterals with Two Pairs of Equal Sides

Image of a rectangle indicating equal opposite sides
A rectangle

 

Rectangles

  • Two pair of equal sides
    • All four sides are not equal.
      Opposite sides are equal.
  • Two pair of equal angles

 

 

Look how a square shape compares to rectangles and rhombi.

The square among a family of rectangles and rhombi. Jacob Rus, CC BY-SA 4.0

The square is the center shape in both lines.

 

 

A parallelogram. Matt Grünewald. CC BY-SA 4.0

Parallelograms

  • Two pair of equal sides. Again, opposite sides are equal.
  • Two pair of equal angles or vertices

 

 

Cross product parallelogram
Different shapes of parallelograms moving through the plane. Source: Fdardel CC-BY-SA 4.0.

Kites, Darts or Deltoids

  • Two pair of equal sides that are next to or adjacent to one another.
  • Two pair of equal angles or vertices
black line drawings of a kite and dart with inscribed circles
On the left, a kite. On the right, a dart or deltoid. Tosha, public domain.

 

The shape on the left is a kite.
Notice that the tip on the top points out.
This is called a convex surface. 

The shape on the right is called a dart or a deltoid.
Notice the tip on the top points inside.
This is called a concave surface.

 

Watch this 2:00 video, Convex and Concave Quadrilaterals by MooMooMath and Science, July 17, 2020.

 

The two lines that are equal are not opposite to each other.  The lines are next to each other.  They are called adjacent lines.  Adjacent means next to or beside.  The top lines that push out and up or in and down are equal adjacent lines.  These lines share a point where they go in opposite directions. This point is the vertex.

The longer lower two lines that form a “V” are also equal adjacent lines.  The lines share the bottom point which is their vertex.

Watch this 6:20 video, Kites in Geometry | Definition and Properties  by Tutors, Feb. 11, 2024.

 

Your Turn

 

Types of Quadrilaterals with One Pair of Parallel Lines

Trapezoid

  • One pair of parallel lines
black line drawing of a trapezoid
A trapezoid with horizontal parallel lines. Public domain.

 

black line drawing of an isosceles trapezoid
A trapezoid with vertical parallel lines. Tomruen, CC BY-SA 4.0.

 

A rotating trapezoid shows that the parallel lines are always parallel no matter the position. Source: A2569875, CC BY-SA 4.0

 

Trapezoids can also be identified by their angles.

 

Your Turn

 

Quadrilaterals that are Irregulars

 

Finally, there is a four-sided shape that does not have equal sides nor any equal angles.  It is not regular with sides or angles.  Therefore, it is called irregular–not regular.
black line drawing of a shape with four unequal sides and four unequal angles.
Irregular quadrilateral also known as a trapezium. Public domain.

Your Turn

 

A Chart about Quadrilaterals

In the left column and the top row, the trapezoids are identified by the size of the inside (interior) angles.  Each of these trapezoids have only one pair of parallel sides/edges.

The remaining four shapes–square, rectangle, parallelogram, and rhombus–are in another group because they each have two pair of parallel sides/edges.

 

A chart with nine quadrilaterals. top row: Acute, right, obtuse center row: Isosceles, rectangle, parallelogram bottom row: 3-sides equal, square, Rhombus The first column and top row are designated as one pair of parallel sides. The middle center and bottom row and middle second row and bottom row are designated as having two parallel sides.
A chart classifying the various quadrilaterals into two groups: 1) with one pair of parallel lines, and 2) with two pair of parallel lines. Tomruen, CC BY-SA 4.0.

Watch this 2:38 video about  Types of Quadrilaterals and their Properties  by MooMooMath and Science, June 1, 2020.

Watch this 2:10 video about What is a Quadrilateral?–Geometric Shapes–Geometry by Socartica, Dec. 21, 2013.

Summary:

  • Identify different types of quadrilaterals

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The Language of Math for Learners of English Copyright © by Lisa Hillyard is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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