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

## Meaning

A square is a polygon the has exactly four sides. (This also way that a square has specifically four vertices, and also exactly four angles.) Discussions the 2-D shapes occasionally refer just to the boundary (the line segments that kind the edge of the figure) or to the interior as well. Once we talk around “dissecting” a parallelogram and rearranging the components to kind a rectangle in stimulate to determine the area that the parallelogram, us are clearly referring come the edges and also the interior. The usual definitions of polygons, however, refer just to the line segments that type the edge of the polygon. Most of the time, context will make clear what friend mean, however you should remain conscious that in some instances you might need to clarify.

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Elementary school curricula generally have youngsters learn the name of special subsets of quadrilateral with specific features. Right here we perform the one-of-a-kind names. Watch the articles on each form for their definitions and special properties.

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

Some K-8 curricula also include

kites (G and, in some definitions H).

## What’s in a word?

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

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

## Classification

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

Like every polygons that have much more than three sides, quadrilaterals have the right to be convex like these , , , or concave choose these , .

Quadrilaterals can be share by even if it is or not their sides, angles, diagonals, or vertices have actually special properties. The group schemes taught in elementary institution involve the variety of pairs that parallel sides, and also the congruence that sides, and also whether or not all the angle are appropriate angles (all angles space congruent). The names of plenty of of these unique quadrilaterals are also typically component of the elementary curriculum, though little else about the properties of these numbers may it is in studied until high school. Elementary school frequently has children learn the name of

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

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

Though the names that are offered to individual figures does no change, the way that they are grouped might depend on the features used to type them. In the group scheme displayed above, parallelograms (B, C, D, E, and also F) have actually a location of their very own (the best hand column), and also even rhombuses (B, C, D) have actually a location (the bottom row), however rectangles (F and B) aren’t identified from the others. In the category scheme below, rectangles (F and also B) have actually the best hand shaft to themselves, but parallelograms room not group in a method that excludes A, i beg your pardon is no a parallelogram. Children in primary grades often find it difficult to entrust anything (geometrical or otherwise) all at once to 2 categories.<1> Casual language also treats shape-names as “exclusive” quite than “inclusive.” Thus, casual language treats square and rectangle as distinct, fairly than dealing with square as a special sort of rectangle, as mathematics does. Similarly, students tend to act rectangles and parallelograms together disjoint classes, quite than seeing a rectangle together a special kind of parallelograms.

Given 6 toy horses and 4 toy cows, and asked whether there are more horses or more animals, really young children often prize “more horses” because in classifying the toys as “horses,” they, for the moment, exclude them together “animals,” even though, if independently asked whether steeds are animals, they will say yes.

Another possible way to divide quadrilaterals is by assessing their diagonals. This might be accessible for center grade student who have learned about perpendicular lines and bisectors.

DiagonalsPerpendicularNot PerpendicularCongruentNot Congruent
BisectorNot BisectorBisectorNot Bisector
SquareSpecial KiteRectangleIsosceles Trapezoid

## Mathematical background

### Properties

The amount of the inner angles in a quadrilateral is 360°. College student who know the analogous an outcome for triangles have the right to convince themselves of this by cut a quadrilateral right into two triangles by illustration a diagonal: each triangle has 180° of edge measure, for this reason the 2 triangles save 360°.

Children can also experiment with this idea by

coloring or labeling the corners that a quadrilateral ,cutting the corners turn off  , andfitting the colored vertices together to present that they every fit snugly approximately a point. Tesselation: The truth that the four vertices right snugly about a solitary point allows us to arrange four copies of a quadrilateral roughly a point. Regardless of the quadrilateral one starts with, four duplicates of it can be i ordered it to to the right snugly about a solitary point. Multiple copies of that foursome will certainly tile the plane. Even if one starts through a concave quadrilateral prefer this , one deserve to group four identical copies of castle snugly roughly a point , and can tile the entire plane with multiple copies. ### Special object (rarely in primary school or center school)

Cyclic quadrilaterals: For part quadrilaterals, that is feasible to pass a solitary circle with all four of that vertices. These one-of-a-kind cases, referred to as cyclic quadrilaterals, incorporate rectangles (and because of this squares) and isosceles trapezoids, but also other shapes that have no special name of their own. Cyclic quadrilaterals have actually the special home that the sum of their opposite angle is a directly angle, or 180 degrees. The implication functions in the other direction, too: any quadrilateral whose opposite angles add up to 180 levels is a cyclic quadrilateral. When the quadrilateral and also the circle passing with its vertices space both shown, the quadrilateral is claimed to be inscribed in ~ the circle and the one is stated to it is in circumscribed around the quadrilateral.  Parallelograms that are not additionally rectangles cannot be inscribed in a circle: they room not cyclic quadrilaterals. A circle the passes through three of the vertices is one of two people too huge to pass v the 4th (blue circle at the left) or too tiny to pass v the 4th (red circle in ~ the right).

In addition to circumscribing circles about a quadrilateral (drawing circles about a quadrilateral, touching each vertex), that is sometimes feasible to inscribe a circle (draw a circle within a quadrilateral so the each side of the quadrilateral is tangent to the circle). This group of quadrilaterals has actually no distinct name that its own, yet includes kites, rhombuses, and also squares in addition to other quadrilateral that have actually no details name.

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Investigations: when investigating one-of-a-kind quadrilaterals and their properties, students find many ways to identify quadrilaterals. Some fertile explorations asking students to look for special properties of angle (congruent or supplementary), sides (parallel, perpendicular, or congruent), and diagonals (perpendicular, bisecting, or congruent).