A square is a polygon that has specifically four sides. (This also method that a quadrilateral has specifically four vertices, and exactly 4 angles.)

## Meaning

A quadrilateral is a polygon that has precisely four sides. (This also way that a square has precisely four vertices, and also exactly 4 angles.)

Discussions of 2-D shapes occasionally refer only to the border (the line segments that type the edges of the figure) or to the internal as well. As soon as we talk about “dissecting” a parallelogram and also rearranging the parts to type a rectangle in stimulate to identify the area the the parallelogram, us are plainly referring come the edges and also the interior. The usual meanings of polygons, however, refer just to the heat segments that kind the edge of the polygon. Most of the time, context will certainly make clear what friend mean, but you must remain mindful that in some situations you might need come clarify.

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Elementary institution curricula typically have kids learn the name of one-of-a-kind subsets of quadrilaterals with certain features. Here we list the one-of-a-kind names. Check out the articles on each form for their definitions and special properties.

trapezoids (A and J are “typical” examples, yet all parallelograms likewise fit the meaning of trapezoids);parallelograms (E is the “typical” example, but all rectangles and also rhombuses also fit the meaning of parallelograms);rectangles (F is the “typical” example, but all squares likewise fit the an interpretation of rectangles);rhombuses (C and also D are the “typical” examples, however all squares additionally fit the definition of rhombuses);squares (B), the many special of them all.

Some K-8 curricula likewise include

kites (G and, in some interpretations H).

## What’s in a word?

quadri- (four) + -lateral (side) way “four-sided.”

Compare quadri- come Spanish cuatro. Lateral method “side” (think of football, for example).

## Classification

Just together triangles and also quadrilaterals space special species of polygons, there are numerous subclasses the quadrilaterals.

Like every polygons the have an ext than 3 sides, quadrilaterals have the right to be convex like these

,
,
,
or concave choose these
,
.

Quadrilaterals deserve to be divide by whether or not their sides, angles, diagonals, or vertices have actually special properties. The group schemes teach in elementary school involve the variety of pairs of parallel sides, and also the congruence that sides, and also whether or not all the angles are ideal angles (all angles room congruent).

The names of numerous of these special quadrilaterals are also typically component of the elementary school curriculum, though little else around the nature of these numbers may it is in studied until high school. Primary school school frequently has kids learn the names of

trapezoids (A and also J space “typical” examples, yet all parallelograms likewise fit the definition of trapezoids);parallelograms (E is the “typical” example, yet all rectangles and rhombuses also fit the definition of parallelograms);rectangles (F is the “typical” example, but all squares likewise fit the meaning of rectangles);rhombuses (C and also D room the “typical” examples, but all squares likewise fit the meaning of rhombuses);squares (B), the most special of lock all; and also sometimeskites (G and also some incorporate H).

The square is additionally the name of the continual quadrilateral — one in which every sides room congruent and also all angles are congruent.

Though the names that are given to individual numbers does not change, the means that they are grouped may depend top top the characteristics used to type them. In the classification scheme shown above, parallelograms (B, C, D, E, and also F) have actually a location of their very own (the appropriate hand column), and even rhombuses (B, C, D) have a location (the bottom row), yet rectangles (F and also B) aren’t differentiated from the others. In the group scheme below, rectangles (F and also B) have the best hand shaft to themselves, yet parallelograms are not group in a means that excludes A, i beg your pardon is no a parallelogram.

Children in primary qualities often discover it difficult to entrust anything (geometrical or otherwise) at the same time to two categories.<1> Casual language additionally treats shape-names as “exclusive” fairly than “inclusive.” Thus, casual language treats square and also rectangle together distinct, fairly than treating square together a special kind of rectangle, as math does. Similarly, students tend to treat rectangles and also parallelograms together disjoint classes, quite than seeing a rectangle together a special form of parallelograms.

Given 6 toy horses and also 4 toy cows, and asked whether over there are an ext horses or an ext animals, really young kids often price “more horses” since in classifying the toys as “horses,” they, for the moment, to exclude, them together “animals,” even though, if separately asked whether horses are animals, they will certainly say yes.

Another possible way to share quadrilaterals is by examining their diagonals. This might be available for middle grade college student who have learned around perpendicular lines and also bisectors.

DiagonalsPerpendicularNot PerpendicularCongruentNot Congruent
BisectorNot BisectorBisectorNot Bisector
SquareSpecial KiteRectangleIsosceles Trapezoid

## Mathematical background

### Properties

The amount of the inner angles in a square is 360°. Student who recognize the analogous an outcome for triangles deserve to convince us of this by cutting a quadrilateral into two triangles by drawing a diagonal: every triangle includes 180° of angle measure, for this reason the two triangles save 360°.

Children can likewise experiment through this idea by

coloring or labeling the corners that a quadrilateral
,cutting the corners off
, andfitting the colored vertices with each other to show that they every fit snugly approximately a point.

Tesselation: The fact that the 4 vertices to the right snugly roughly a solitary point permits us to arrange four copies of a quadrilateral about a point. Nevertheless of the square one beginning with, four copies of it deserve to be arranged to to the right snugly roughly a single point.

Multiple copies of the foursome will tile the plane.
Even if one starts with a concave quadrilateral choose this
, one can group four identical duplicates of castle snugly around a allude
, and also can brick the entire plane with many copies.

### Special topics (rarely in elementary school or middle school)

Cyclic quadrilaterals: For part quadrilaterals, that is feasible to happen a single circle through all 4 of the vertices. These unique cases, called cyclic quadrilaterals, include rectangles (and thus squares) and isosceles trapezoids, but also other forms that have no one-of-a-kind name of your own. Cyclic quadrilaterals have actually the special home that the sum of your opposite angles is a right angle, or 180 degrees. The implication works in the other direction, too: any kind of quadrilateral whose opposite angles add up come 180 levels is a cyclic quadrilateral.

When the quadrilateral and the circle passing through its vertices space both shown, the square is claimed to it is in inscribed in ~ the circle and also the one is claimed to it is in circumscribed around the quadrilateral.

Parallelograms that space not likewise rectangles cannot be inscriptions in a circle: they room not cyclic quadrilaterals. A circle the passes through three of the vertices is either too huge to pass with the fourth (blue circle in ~ the left) or too little to pass with the fourth (red circle at the right).

In enhancement to circumscribing circles roughly a square (drawing circles roughly a quadrilateral, poignant each vertex), that is sometimes possible to inscribe a one (draw a circle within a square so the each next of the square is tangent to the circle). This team of quadrilaterals has actually no special name the its own, yet includes kites, rhombuses, and also squares together with other quadrilaterals that have no particular name.

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Investigations: when investigating one-of-a-kind quadrilaterals and their properties, student find countless ways to identify quadrilaterals. Some fertile explorations asking students come look for one-of-a-kind properties of angles (congruent or supplementary), political parties (parallel, perpendicular, or congruent), and diagonals (perpendicular, bisecting, or congruent).