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

## Meaning

A quadrilateral is a polygon the has precisely four sides. (This also way that a square has specifically four vertices, and also exactly four angles.) Discussions the 2-D shapes periodically refer only to the boundary (the heat segments that type the edges of the figure) or come the interior as well. Once we talk about “dissecting” a parallelogram and rearranging the components to type a rectangle in order to recognize the area of the parallelogram, us are clearly referring come the edges and also the interior. The usual interpretations of polygons, however, refer just to the line segments that kind the edge of the polygon. Many of the time, context will make clean what you mean, yet you have to remain mindful that in some cases you might need come clarify.

You are watching: A quadrilateral with 4 square corners

Elementary institution curricula frequently have children learn the names of special subsets of square with certain features. Below we list the unique names. Check out the short articles on each kind for your definitions and also special properties.

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

Some K-8 curricula likewise include

kites (G and, in some definitions H).

## What’s in a word?

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

Compare quadri- come Spanish cuatro. Lateral means “side” (think of football, because that example).

## Classification

Just together triangles and quadrilaterals are special varieties of polygons, over there are many subclasses of quadrilaterals.

Like every polygons the have much more than three sides, quadrilaterals can be convex prefer these , , , or concave like these , .

Quadrilaterals can be classified by whether or no their sides, angles, diagonals, or vertices have special properties. The group schemes taught in elementary institution involve the variety of pairs the parallel sides, and the congruence the sides, and also whether or no all the angles are ideal angles (all angles space congruent). The name of many of these one-of-a-kind quadrilaterals are also typically component of the elementary school curriculum, though little else around the properties of these figures may be studied until high school. Primary school school commonly has kids learn the name of

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

The square is also the surname of the continual quadrilateral — one in which all sides space congruent and all angles are congruent.

Though the names the are provided to individual numbers does no change, the means that they room grouped may depend top top the attributes used to type them. In the classification scheme presented above, parallelograms (B, C, D, E, and F) have a ar of their very own (the ideal hand column), and even rhombuses (B, C, D) have actually a ar (the bottom row), yet rectangles (F and B) aren’t identified from the others. In the classification scheme below, rectangles (F and also B) have actually the ideal hand obelisk to themselves, however parallelograms space not group in a method that excludes A, i beg your pardon is no a parallelogram. Children in primary grades often uncover it tough to assign anything (geometrical or otherwise) concurrently to 2 categories.<1> Casual language additionally treats shape-names as “exclusive” quite than “inclusive.” Thus, casual language treats square and rectangle as distinct, rather than treating square together a special type of rectangle, as math does. Similarly, students have tendency to law rectangles and parallelograms as 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 there are more horses or more animals, really young children often answer “more horses” since in classifying the toys as “horses,” they, for the moment, exclude, them together “animals,” even though, if separately asked whether horses are animals, they will certainly say yes.

Another possible way to divide quadrilaterals is by evaluating their diagonals. This may be easily accessible for middle grade college student who have actually learned about perpendicular lines and bisectors.

DiagonalsPerpendicularNot PerpendicularCongruentNot Congruent
BisectorNot BisectorBisectorNot Bisector
SquareSpecial KiteRectangleIsosceles Trapezoid

## Mathematical background

### Properties

The amount of the internal angles in a square is 360°. College student who recognize the analogous result for triangles deserve to convince us of this by cutting a quadrilateral right into two triangle by drawing a diagonal: every triangle consists of 180° of edge measure, therefore the two triangles contain 360°.

Children can also experiment through this idea by

coloring or labeling the corners the a square ,cutting the corners turn off  , andfitting the fancy vertices with each other to present that they every fit snugly around a point. Tesselation: The fact that the four vertices fit snugly approximately a single point permits us come arrange four copies of a quadrilateral roughly a point. Regardless of the square one beginning with, four duplicates of it have the right to be i ordered it to fit snugly about a solitary point. Multiple duplicates of the foursome will certainly tile the plane. Even if one starts v a concave quadrilateral like this , one can group 4 identical duplicates of castle snugly roughly a suggest , and also can tile the entire airplane with multiple copies. ### Special topics (rarely in elementary school or middle school)

Cyclic quadrilaterals: For part quadrilaterals, the is feasible to happen a solitary circle through all four of that vertices. These unique cases, called cyclic quadrilaterals, encompass rectangles (and because of this squares) and isosceles trapezoids, but likewise other forms that have no one-of-a-kind name of your own. Cyclic quadrilaterals have the special home that the amount of their opposite angle is a directly angle, or 180 degrees. The implication functions in the various other direction, too: any type of quadrilateral who opposite angles include up come 180 degrees is a cyclic quadrilateral. When the quadrilateral and also the circle passing with its vertices room both shown, the square is said to it is in inscribed within the circle and the one is stated to it is in circumscribed around the quadrilateral.  Parallelograms that space not also rectangles cannot be inscriptions in a circle: they room not cyclic quadrilaterals. A circle that passes through 3 of the vertices is either too large to pass with the fourth (blue circle in ~ the left) or too small to pass with the 4th (red circle at the right).

In enhancement to circumscribing circles about a square (drawing circles approximately a quadrilateral, touching each vertex), the is sometimes possible to inscribe a one (draw a circle in ~ a quadrilateral so the each next of the quadrilateral is tangent come the circle). This team of quadrilaterals has actually no special name that its own, yet includes kites, rhombuses, and squares along with other square that have actually no certain name.

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Investigations: when investigating distinct quadrilaterals and also their properties, students find plenty of ways to distinguish quadrilaterals. Some fertile explorations ask students come look for unique properties of angle (congruent or supplementary), political parties (parallel, perpendicular, or congruent), and also diagonals (perpendicular, bisecting, or congruent).