square

• n.

• 平方、二乗、四角

• adj.

• 四角の

関

sq

　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

WordNet

1. raise to the second power
2. (geometry) a plane rectangle with four equal sides and four right angles; a four-sided regular polygon; "you can compute the area of a square if you know the length of its sides" (同)foursquare
3. the product of two equal terms; "nine is the second power of three"; "gravity is inversely proportional to the square of the distance" (同)second power
4. make square; "Square the circle"; "square the wood with a file" (同)square_up
5. something approximating the shape of a square
6. a hand tool consisting of two straight arms at right angles; used to construct or test right angles; "the carpenter who built this room must have lost his square"
7. any artifact having a shape similar to a plane geometric figure with four equal sides and four right angles; "a checkerboard has 64 squares"
8. someone who doesnt understand what is going on (同)lame
9. a formal and conservative person with old-fashioned views (同)square toes
10. rigidly conventional or old-fashioned (同)straight
11. without evasion or compromise; "a square contradiction"; "he is not being as straightforward as it appears" (同)straightforward, straight
12. be compatible with; "one idea squares with another"
13. cause to match, as of ideas or acts
14. having four equal sides and four right angles or forming a right angle; "a square peg in a round hole"; "a square corner"
15. leaving no balance; "my account with you is now all square"
16. pay someone and settle a debt; "I squared with him"
17. position so as to be square; "He squared his shoulders"
18. shaped like a square
19. a steel square used by carpenters; larger than a try square
20. having been made square

PrepTutorEJDIC

1. 『正方形』;四角な物;(チェス・チェッカーなどの盤の,正方形の)目,ます目 / (四角い)『広場』(街路の交差点にあって,しばしば中央に植木や芝などが植えてあり,小公園になっている);《おもに英》広場の回りの建物(街路)(《略》Sq.) / (四方を街路で囲まれた方形の)一区画,ブロック / 直角定規,かね尺 / (数の)『2乗』,平方(《略》sq.) / 《俗》旧式な人 / 『正方形の』,四角な,直角の,直角をなす / 角ばった,がっかりした / 『平方の』,2乗の(《略》『sq.』) / 《補語にのみ用いて》対等の,五分五分の(even);貸し借りにない / 正直な(honest),公正な(fair),正しい(just) / 率直な,はっきりした,きっぱりした(direct) / 実質のある,十分な / 《俗》しゃちほこばった / =squarely / …‘を'正方形(四角)にする;直角にする / …‘を'正方形(四角)に区切る《+『off』+『名』,+『名』+『off』》 / 〈肩・ひじなど〉‘を'張る / (人と)〈勘定〉‘を'決済する,清算する《+『名』〈勘定〉+『with』+『名』〈人〉》 / (…と)…‘を'一致させる,適合させる《+『名』+『with』+『名』》 / 《受動態で》〈数〉‘を'2乗する;〈ある形・図形など〉‘の'平方積(面積)を求める / 〈人〉‘を'買収する,抱き込む,…‘に'わいろを使う / 〈試合の得点〉‘を'同点にする / (…と)一致する,適合する《+『with』+『名』》
2. 四角の,正方形の

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/05/28 10:06:50」(JST)

wiki ja

[Wiki ja表示]

スクウェア、スクエア (square)

• 正方形
  • スクエアストローク：エンジン用語。ボアとストロークが同じ＝正方形であることから。
• 平方（2乗）
• 広場/プラザ#スクエアの例
• 堅物、わからず屋、保守的な人物、エスタブリッシュメント、上流階級の人間
• 商業地域
• ゴルフに於いては飛球線と90度正対していること。
• 天体の黄経差が90度であること。矩。
• 直角定規（スコヤ）
• 100平方フィート（約9.290平方メートル）に等しい面積の単位 square (unit)
• 番号記号 (#)

名称

• スクウェア (ゲーム会社) - ゲーム製作会社。スクウェア・エニックスの前身。
• スクウェア (アダルトゲーム会社) - 「U・Me SOFT」等のブランドを持つアダルトゲーム会社。
• T-SQUARE（T-スクエア） - 日本のフュージョンバンド。
• スクエア (都市伝説) - 物語のひとつ。
• スクエア (劇団) - 日本の劇団。
• ザ・スクエア - 聖蹟桜ヶ丘駅前にあるショッピングセンター。
• ジャンプスクエア - 雑誌。
• アメージングスクエア - 東京都足立区に所在するスポーツ施設。
• SQuaRE(Software Product Quality Requirements and Evaluation) - ISO/IEC 25000シリーズ
• Square, Inc. - アメリカ合衆国のモバイル決済企業

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wiki en

[Wiki en表示]

For other uses, see Square (disambiguation).

Square
A regular quadrilateral (tetragon)
Type	Regular polygon
Edges and vertices	4
Schläfli symbol	{4} t{2}
Coxeter diagram
Symmetry group	Dihedral (D4), order 2×4
Internal angle (degrees)	90°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.

A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.

Characterizations

The square has dihedral point symmetry, Dih₄, or orbifold (*44), with reflective subsymmetries: *22, *, and rotational subsymmetries: 44, 22, 1. These subsymmetries can be seen in 7 lower symmetry quadrilaterals, depending on whether the mirror lines are on the vertices or edges. The gyration square has the full geometry of the square, but has edge markings that define a rotating orientation of edges.

A convex quadrilateral is a square if and only if it is any one of the following:

• a rectangle with two adjacent equal sides
• a quadrilateral with four equal sides and four right angles
• a parallelogram with one right angle and two adjacent equal sides
• a rhombus with a right angle
• a rhombus with all angles equal
• a quadrilateral where the diagonals are equal and are the perpendicular bisectors of each other, i.e. a rhombus with equal diagonals

Perimeter and area

The area of a square is the product of the length of its sides.

The perimeter of a square whose four sides have length is

and the area A is

In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.

The area can also be calculated using the diagonal d according to

In terms of the circumradius R, the area of a square is

and in terms of the inradius r, its area is

A convex quadrilateral with successive sides a, b, c, d is a square if and only if

Coordinates and equations

plotted on Cartesian coordinates.

The coordinates for the vertices of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (x_i, y_i) with −1 < x_i < 1 and −1 < y_i < 1. The equation

specifies the boundary of this square. This equation means "x² or y², whichever is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and equals . Then the circumcircle has the equation

Alternatively the equation

can also be used to describe the boundary of a square with center coordinates (a, b) and a horizontal or vertical radius of r.

Construction

Construction of a square using a compass and straightedge.

The animation at the right shows how to construct a square using a compass and straightedge.

Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a parallelogram (opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles) and therefore has all the properties of all these shapes, namely:

• The diagonals of a square bisect each other and meet at 90°
• The diagonals of a square bisect its angles.
• Opposite sides of a square are both parallel and equal in length.
• All four angles of a square are equal. (Each is 360°/4 = 90°, so every angle of a square is a right angle.)
• All four sides of a square are equal.
• The diagonals of a square are equal.

Other facts

• The diagonals of a square are (about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant, was the first number proven to be irrational.
• A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
• If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
• If a circle is circumscribed around a square, the area of the circle is (about 1.5708) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is (about 0.7854) times the area of the square.
• A square has a larger area than any other quadrilateral with the same perimeter.
• A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
• The square is in two families of polytopes in two dimensions: hypercube and the cross polytope. The Schläfli symbol for the square is {4}.
• The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D₄.
• If the inscribed circle of a square ABCD has tangency points E on AB, F on BC, G on CD, and H on DA, then for any point P on the inscribed circle,

Squares inscribed in triangles

Main article: Triangle § Squares

Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.

The fraction of the triangle's area that is filled by the square is no more than 1/2.

Squaring the circle

Squaring the circle is the problem, proposed by ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients.

Non-Euclidean geometry

In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.

In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles.

In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles.

Examples:

Two squares can tile the sphere with 2 squares around each vertex and 180-degree internal angles. Each square covers an entire hemisphere and their vertices lie along a great circle. This is called a spherical square dihedron. The Schläfli symbol is {4,2}.

Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.

Squares can tile the Euclidean plane with 4 around each vertex, with each square having an internal angle of 90°. The Schläfli symbol is {4,4}.

Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72-degree internal angles. The Schläfli symbol is {4,5}. In fact, for any n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.

Crossed square

A crossed square is a faceting of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih₂, order 4. It has the same vertex arrangement as the square, and is vertex-transitive. It appears as two 45-45-90 triangle with a common vertex, but the geometric intersection is not considered a vertex.

A crossed square is sometimes likened to a bow tie or butterfly. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals.

The interior of a crossed square can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.

A square and a crossed square have the following properties in common:

• Opposite sides are equal in length.
• The two diagonals are equal in length.
• It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

It exists in the vertex figure of a uniform star polyhedra, the tetrahemihexahedron:

Graphs

3-simplex (3D)

The K₄ complete graph is often drawn as a square with all 6 edges connected. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).

See also

• Cube
• Pythagorean theorem
• Square lattice
• Squircle
• Unit square

References

External links

• Animated course (Construction, Circumference, Area)
• Weisstein, Eric W., "Square", MathWorld.
• Definition and properties of a square With interactive applet
• Animated applet illustrating the area of a square

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★リンクテーブル★

リンク元	「平方」「二乗」「四角」
拡張検索	「least-squares analysis」「least squares」「least-squares method」「square wave」

「平方」

　　[★]

英

• square、sq

関

四角、二乗

「二乗」

　　[★]

英

square

関

四角、平方

「四角」

　　[★]

英

square

関

平方、二乗

「least-squares analysis」

　　[★]

最小二乗法

関

least squares、least-squares method、Rietveld refinement

「least squares」

　　[★]

最小二乗、最小自乗

関

least-squares analysis、Rietveld refinement

「least-squares method」

　　[★]

• 最小二乗法

関

least-squares analysis

「square wave」

　　[★]

• 方形波

関

square pulse