What is square in shape?

square, in geometry, a plane figure with four equal sides and four right (90°) angles. A square is a special kind of rectangle (an equilateral one) and a special kind of parallelogram (an equilateral and equiangular one).

• Floor and Wall Tiles.
• Paper Napkins.
• Chess Board.
• Stamps.
• Cushions.
• Clock.
• Bread.
• Cheese Slice.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ◻ {\displaystyle \square } ABCD.[1]

A convex quadrilateral is a square if and only if it is any one of the following:[2][3]

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all 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:[5]

The perimeter of a square whose four sides have length ℓ {\displaystyle \ell } is

and the area A is

Since four squared equals sixteen, a four by four square has an area equal to its perimeter. The only other quadrilateral with such a property is that of a three by six rectangle.

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

since the area of the circle is π R 2 , {\displaystyle \pi R^{2},} the square fills 2 / π ≈ 0.6366 {\displaystyle 2/\pi \approx 0.6366} of its circumscribed circle.

In terms of the inradius r, the area of the square is

hence the area of the inscribed circle is π / 4 ≈ 0.7854 {\displaystyle \pi /4\approx 0.7854} of that of the square.

Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:

with equality if and only if the quadrilateral is a square.

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 (xi, yi) with −1 < xi < 1 and −1 < yi < 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 is equal to 2 . {\displaystyle {\sqrt {2}}.} 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. The square is therefore the shape of a topological ball according to the L1 distance metric.

The following animations show how to construct a square using a compass and straightedge. This is possible as 4 = 22, a power of two.

The square has Dih4 symmetry, order 8. There are 2 dihedral subgroups: Dih2, Dih1, and 3 cyclic subgroups: Z4, Z2, and Z1.

A square is a special case of many lower symmetry quadrilaterals:

These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order.[11]

Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilaterals. r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a rectangle, and p4 is the symmetry of a rhombus. These two forms are duals of each other, and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid, and p2 is the symmetry of a kite. g2 defines the geometry of a parallelogram.

Only the g4 subgroup has no degrees of freedom, but can seen as a square with directed edges.

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, proposed by ancient geometers, is the problem 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.

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:

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.[12]

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:

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

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

The other properties of the square such as area and perimeter also differ from that of a rectangle. Let us learn here in detail, what is a square and its properties along with solved examples.

Square is a regular quadrilateral, which has all the four sides of equal length and all four angles are also equal. The angles of the square are at right-angle or equal to 90-degrees. Also, the diagonals of the square are equal and bisect each other at 90 degrees.

A square can also be defined as a rectangle where two opposite sides have equal length.

The above figure represents a square where all the sides are equal and each angle equals 90 degrees.

Just like a rectangle, we can also consider a rhombus (which is also a convex quadrilateral and has all four sides equal), as a square, if it has a right vertex angle.

In the same way, a parallelogram with all its two adjacent equal sides and one right vertex angle is a square.

Also, read:

A square is a four-sided polygon which has it’s all sides equal in length and the measure of the angles are 90 degrees. The shape of the square is such as, if it is cut by a plane from the center, then both the halves are symmetrical. Each half of the square then looks like a rectangle with opposite sides equal.

The most important properties of a square are listed below:

The area and perimeter are two main properties that define a square as a square. Let us learn them one by one:

Area of the square is the region covered by it in a two-dimensional plane. The area here is equal to the square of the sides or side squared. It is measured in square unit.

Area = side2 per square unit

If ‘a’ is the length of the side of square, then;

Area = a2 sq.unit

Also, learn to find Area Of Square Using Diagonals.

The perimeter of the square is equal to the sum of all its four sides. The unit of the perimeter remains the same as that of side-length of square.

Perimeter = Side + Side + Side + Side = 4 Side

Perimeter = 4 × side of the square

If ‘a’ is the length of side of square, then perimeter is:

Perimeter = 4a unit

The length of the diagonals of the square is equal to s√2, where s is the side of the square. As we know, the length of the diagonals is equal to each other. Therefore, by Pythagoras theorem, we can say, diagonal is the hypotenuse and the two sides of the triangle formed by diagonal of the square, are perpendicular and base.

Since, Hypotenuse2 = Base2 + Perpendicular2

Hence, Diagonal2 = Side2 + Side2

\(\begin{array}{l} Diagonal = \sqrt{2side^2}\end{array} \)

Hence, d = s√2

Where d is the length of the diagonal of a square and s is the side of the square.

Diagonal of square is a line segment that connects two opposite vertices of the square. As we have four vertices of a square, thus we can have two diagonals within a square. Diagonals of the square are always greater than its sides.

Below given are some important relation of diagonal of a square and other terms related to the square.

Problem 1: Let a square have side equal to 6 cm. Find out its area, perimeter and length of diagonal.

Solution: Given, side of the square, s = 6 cm

Area of the square = s2 = 62 = 36 cm2

Perimeter of the square = 4 ×  s = 4 × 6 cm = 24cm

Length of the diagonal of square = s√2 = 6 × 1.414 = 8.484

Problem 2: If the area of the square is 16 sq.cm., then what is the length of its sides. Also find the perimeter of square.

Solution: Given, Area of square = 16 sq.cm.

As we know,

area of square =side2

Therefore, by substituting the value of area, we get;

16 = side2

side = √16 = √(4×4) = 4 cm

Hence, the length of the side of square is 4 cm.

Now, the perimeter of square is:

P = 4 x side = 4 x 4 = 16 cm.

Learn more about different geometrical figures here at BYJU’S. Also, download its app to get a visual of such figures and understand the concepts in a better and creative way.

A square is a quadrilateral geometric shape that consists of four equal sides and four equal angles. The area of a square can be obtained by evaluating the numerical product of its two sides. The perimeter of the square is four times the magnitude of the side.

1. The sum of all the angles of a quadrilateral is equal to 360°, and all the angles of a square are equal in magnitude; therefore, the angles formed between the two adjacent sides of a square is equal to 90°.

2. The opposite sides of a square are parallel to each other.

3. The two diagonals formed by joining the opposite corners of a square are equal in length and have the value √2 times the magnitude of the side of the square.

4. The diagonals bisect each other at an angle of 90°.

Most of the tiles used in constructing and decorating home walls and floors are square-shaped. Hence, the tiles are one of the prominent examples of square-shaped objects used in everyday life.

Everyone has a stack of napkins placed on the top of the dining table. A napkin is a square-shaped piece of paper or a cloth that is used to wipe the mouth, hands, or objects. Hence, it is yet another example of square-shaped objects used in daily life.

The chessboard is one of the best examples of the square-shaped objects used in everyday life. Not only the outer boundary of the chessboard is shaped like a square, but it also contains 64 small square boxes on the inside.

A stamp is a small piece of paper that is affixed on the front side of the envelope containing the letter before posting it. Most of the stamps are square in shape.

A cushion is a bag of fabric that is stuffed with cotton, fur, wool, or beans. The cushions used to decorate the living room are generally square in shape.

In real life, the square geometric shape can be observed easily by looking at the front face of a cubical desk clock. Some of the wall clocks are also square-shaped.

A loaf of bread is generally cuboidal in shape; however, if you pick one thin slice of the bread you can easily identify its square shape.

The shape of the cheese slices is square. This is because it is easier for the manufacturing industries to wrap a square slice instead of wrapping a round slice.

Most of the windows installed in homes are square in shape. In some cases, not just the outer frame but the glass inside the grilles is also square-shaped.

If you clearly observe the front face of the chocolate cube, all four sides are equal, and the opposite sides are parallel to each other. Hence, the cube taken out of a bar of chocolate is yet another example of the square-shaped objects seen in everyday life.

Photo frames come in a variety of shapes. One of the most popular shapes of a photo frame is a square.

Some of the biscuits are baked in the shape of a quadrilateral that has all sides equal, i.e., in the shape of a square. So, next time while eating a biscuit don’t forget to get yourself reminded of the square geometric shape and its properties.

A craft paper is a colourful and textured piece of paper that is used to model a number of paper artefacts. It is generally square in shape.

One of the most commonly used objects in daily life that is square in shape is a bedsheet. It thoroughly covers the mattress and protects it from dust and stains.

Take a look at the images given below. You might have come across objects like a photo frame, or a craft paper in day-to-day life. Can you identify what is common in them?

All of them have a square shape.

A square is a regular polygon having four equal sides and equal angles that measure 90° each.

A square is a two-dimensional closed shape with 4 equal sides and 4 vertices. Its opposite sides are parallel to each other. We can also think of a square as a rectangle with equal length and breadth.

Looking around, you can find many things that resemble the square shape. Common examples of this shape include a chessboard, craft papers, bread slice, photo frame, pizza box, a wall clock, etc.

Area represents space occupied by a shape or figure whereas perimeter is the length of the outer boundary of the shape. Let’s discuss the formula for finding the area and perimeter of a square.

The area of a two-dimensional shape is defined as the amount of space covered by the shape if we were to keep it on a flat table.

For a square of side length “s” units, the area is given by the formula:

Area $= \text{side} \times \text{side} = \text{S}^2$

The area is expressed in square units, such as $\text{cm}^2$, $\text{cm}^2$, etc.

The perimeter of a two-dimensional shape is defined as the total length of its boundary.

For a square of side length “s” units, the perimeter is given by the formula:

Perimeter $= \text{side} + \text{side} + \text{side} + \text{side} = 4$ $\text{x}$ $\text{s}$

The perimeter is expressed in linear units, such as cm, inches, m, etc.

Example 1: The side of a square paper is 12 feet. Find the area of the paper.

Solution:

We know that the area of a square is given by $\text{s}^2$, where $\text{s} =$ length of the side.

For the given square, s $= 12$ feet

Therefore, the area of the square paper is given by:

Area $= \text{s}^2 = 12 \times 12 = 144$ sq. ft.

Example 2: If the perimeter of a square measures 68 cm, what is the measure of its side?

Solution:

We know that the perimeter of a square is given by 4 x side.

It is given that the perimeter is 68 cm.

Therefore, 4 x side $= 68$

Which means, side $= \frac{68}{4} = 17$ cm

Example 3: What is the perimeter of a square that has a side of 15 meters?

Solution: We know that the perimeter of a square is given by 4 x s, where s represents the length of each side.

It is given that the side s $= 15$ meters.

Therefore, perimeter $= 4 \times 15 = 60$ meters.