Square

Square
Square
Ribs	four
Shlefly Symbol	{four}
Type of symmetry	Dihedral group (D 4 )
Square	t 2
Inside corner	90 °

A square is a regular quadrilateral , that is, a quadrilateral in which all angles are equal and all sides are equal. A square is at the same time a special case of a rhombus and a rectangle .

Square

Square Properties

The lengths of all sides are equal.
All corners of the square are straight.
The diagonals of a square are equal, mutually perpendicular, the intersection point is divided in half and are bisectors of angles.

Properties

Let be $t$ {\ displaystyle t} - side of a square, $R$ {\ displaystyle R} - radius of the circumscribed circle , $r$ {\ displaystyle r} - radius of the inscribed circle . Then the center of the circumscribed and inscribed circles of the square coincides with the intersection point of its diagonals, and
- the radius of the inscribed circle of the square is equal to half the side of the square:
  $r={\frac {t}{2}}$ ${\ displaystyle r = {\ frac {t} {2}}}$ $r={\frac {t}{2}}$ ,
- the radius of the circumscribed circle of the square is equal to half the diagonal of the square:
  $R={\frac {\sqrt {2}}{2}}t$ ${\ displaystyle R = {\ frac {\ sqrt {2}} {2}} t}$ $R={\frac {\sqrt {2}}{2}}t$ ,
A square has the greatest symmetry among all quadrangles. He has:
- one axis of symmetry of the fourth order (the axis perpendicular to the plane of the square and passing through its center);
- four axes of symmetry of the second order (which for a flat figure is equivalent to reflections), of which two pass along the diagonals of the square, and the other two parallel to the sides.
The diagonals of the square are equal, mutually perpendicular, the intersection point is divided in half and divide the corners of the square in half.

The area of the square and the length of its sides

Lines of symmetry

Square Perimeter

- the perimeter of the square is:
  $P=4t=4{\sqrt {2}}R=8r$ ${\ displaystyle P = 4t = 4 {\ sqrt {2}} R = 8r}$ ,

Square Area

Here are formulas that are specific to a square. See also formulas for the area of arbitrary quadrangles .

- square $S$ ${\ displaystyle S}$ square equal
  $S=t^{2}=2R^{2}=4r^{2}$ ${\ displaystyle S = t ^ {2} = 2R ^ {2} = 4r ^ {2}}$ .

Non-Euclidean geometry

In non-Euclidean geometry, a square (more broadly) is a polygon with four equal sides and equal angles.

Building a square using a compass and a ruler

Folding a square from an arbitrary piece of paper

Variety of Squares

The squares are the faces of the cube - one of the five regular polyhedra .

Graphs: K _{4 a} complete graph is often depicted as a square with six edges.

3- simplex (3D)

A chessboard has the shape of a square and is divided into 64 squares of two colors. The square board for international drafts is divided into 100 squares of two colors. The boxing ring and the square are square .

The square flag of Lima is divided into two black and two yellow squares, being raised on a ship in the harbor , means that the ship is in quarantine .

Links

A square, a geometric figure // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.

Square

Square Properties

Properties

Square Perimeter

Square Area

Non-Euclidean geometry

Variety of Squares

See also

Links

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