Square Meaning
In Geometry, a square is a two-dimensional two-dimensional figure with four equal sides and every one the four angles are adequate to 90 degrees. The properties of a rectangle are somewhat similar to a square, but the difference between the two is, a rectangle has only its opposite sides equal. Therefore, a rectangle can be said to be a square as long as all its four sides are of equal length.
The Square
Number of sides = 4
Number of vertices = 4
Area = Side * Side
Perimeter of square formula = 4(Side)
The different properties for the square like the area and the perimeter of square formulae also differ from that of the rectangle. Let’s learn in detail about what a square is and its different properties.
Definition
Square may be a regular quadrilateral, which has all the four sides of equal length and every one of the four angles also are equal. The angles of the square are at right-angle or 90-degrees. The diagonals for the square are equal and also they bisect each other at 90 degrees.
A square also can be defined as a rectangle where two opposite sides have equal length.
(Image to be added soon)
The above figure represents a square where all the edges are equal and every angle equals 90 degrees.
Just like a rectangle, we will also consider a rhombus (which is additionally a convex quadrilateral and has all four sides equal), to be a square, if it's a right vertex angle.
In the very same way, the parallelogram with all of its two adjacent and equal sides and one right vertex angle is a square.
Shape of Square
A square may be a four-sided polygon which has its all sides equally long and therefore the measure of the angles are 90 degrees. The shape of the square is like if it's cut by a plane from the middle, then both the halves are symmetrical. Each half the square then seems like a rectangle with opposite sides equal.
Properties of a Square
The important properties of a square are mentioned below:
All four interior angles are equal to 90°
All four sides of the square are congruent or equal to each other
The opposite sides of the square are parallel to each other
The diagonals of the square bisect one another at 90°
The two diagonals of the square are adequate to one another
The square has 4 vertices and 4 sides
The diagonal for the square when divided, comes out as two similar isosceles triangles
The length of the diagonals is greater than that of the sides for the square
Area and Perimeter of Square Formula
The area and perimeter are two main properties that outline a square as a square. Let us learn them one by one:
Area
Area of the square is the region covered by it during a two-dimensional plane. The area here is adequate to the square of the edges or side squared. It is measured in square units.
Area = side2 per square unit
If ‘a’ is that the length of the side of the square, then;
Area = a2 sq.unit
Also, learn to seek out the Area of Square using Diagonals.
Perimeter
The perimeter of the square is adequate to the sum of all its four sides. The unit of the perimeter remains the same as that of the side-length of the square.
Perimeter equals Side + Side + Side + Side equals 4 Side
Perimeter = 4 × side of the square
If ‘a’ is that the length of the side of the square, then perimeter is: Perimeter = 4a unit.
Length of Diagonal of a Square
The length for the diagonals of the square equals to s√2, where s is the side for the square. As the length for the diagonals is equal to one another, by Pythagoras theorem, we can say, the diagonal is the hypotenuse and the two sides of the triangle formed by the diagonal of the square, are perpendicular and base.
Since, Hypotenuse2 = Base2 + Perpendicular2
Hence, Diagonal2 = side2 + side2
Diagonal = √2side2
d = s√2
Where d is that the length of the diagonal of a square and s is that the side of the square.
Below are some Amazing Facts about a Square :
A square has 4 lines of reflectional symmetry, two lines on midpoints and two through the opposite vertices.
It's all sides are congruent and all angles equal to 90 degrees.
If the perimeter is kept the same for all the quadrilaterals, the square has the largest area in all of them.
The square can be defined as a specialized case of a rectangle, but the rectangle can not be defined in terms of a square.
The 3-D View of a Square : Cube
The square is a 2 dimensional shape whereas the cube has a three-dimensional solid shape. The cube has 6 faces, 12 edges, and 8 vertices. Also, the cube has all faces equal to the squares of the same length of side. Following are the points about a cube
It's volume is calculated by (side)³. Here, sides mainly consist of length, breadth, height. The volume here is in cubic units.
When a cube is opened, it gives 6 squares. 4 faces for the 4 sides , one for top and the last for bottom.
Examples of objects that are cube shaped include building, dice, box etc.
FAQs on Square
1. What is the formula of the perimeter of a square?
As all the sides of a square are equal, we only need one side to find its perimeter. The perimeter for the given square is written as a + a + a + a equals 4a units. Therefore, the formula for the perimeter of the square equals 4 × (length of any one of the sides). By this formula, we can also find the length of any one of the sides when we divide the parameter of the square by 4.
2. What is the area of a square?
The area is measured in "square" units. The area of a figure is the number of squares required to hide it completely, like tiles on a floor. Area of a square= side times side. As all the sides for the square are the same, it can be simply the length for 1 side squared. The area of a square can also be represented in the form of the volume and in terms of the 'diagonal' of that square. But, there is no direct relation between the perimeter and the area of a square.
3. What are the properties of a square?
The diagonals of a square bisect one another and meet at 90°.
The diagonals of a square bisect its angles. Diagonals of a square are equal. Also, whenever the diagonal bisects the angles of the square, each angle measures 45°. Opposite sides of a square are both parallel and equally long.
All of the four angles for the square are the same (each of them being 360°/4 equals 90°, which is a right angle). All four sides of a square are equal.
4. What is the diagonal of a square?
Diagonal of a square may be a line segment that connects two opposite vertices of the square. As we've four vertices of a square, thus we will have two diagonals within a square. Diagonals for the square are always greater than the sides. The area of a square in terms of its diagonal is d²/2. Here, d is the length of the diagonal of that square. Similarly, diagonals can also be used to calculate the perimeter of a square.
(i) Show that if the and bisect each other at right angles, then it is a square.
Suppose a quadrilateral ABCD, where the diagonals AC and BD intersect each other at O.
Given :
Diagonals of a quadrilateral are equal.
They bisect each other at right angles.
Therefore, if ABCD is a square, it should be a parallelogram where AB = BC = CD = AD and one of its interior angles is 90 degree.
So, in triangle ΔAOB and ΔCOD:
AO = CO ( because Diagonals bisect each other)
OB = OD ( because Diagonals bisect each other)
∠AOB = ∠COD (Vertically opposite angles are equal)
So, by SAS ( Side angle side) congruence rule, ΔAOB ≅ ΔCOD.
Hence, by CPCT (Corresponding parts of congruent triangles):
AB = CD { Equation (i) }
∠OAB = ∠OCD.
However, for line AB and CD, these are alternate interior angles and alternate interior angles are equal to each other when the two of the lines are parallel.
So, AB || CD { Equation (ii)}
and from equation (i) and (ii), we have ABCD is a parallelogram.
Now, in ΔAOD and ΔCOD :
AO = CO (Diagonals of the parallelogram bisect each other)
∠AOD = ∠COD ( It's given in question that each is 90º)
OD = OD (Common side)
Therefore, by SAS (side angle side) congruence rule:
ΔAOD ≅ ΔCOD
So, AD = DC { Equation (iii) }
Also, we know that, AD = BC and AB = CD
( Opposite sides of parallelogram ABCD are equal).
By this, we get, AB = BC = CD = DA.
Hence, all the sides of quadrilateral ABCD are equal to each other.
Also, In ΔADC and ΔBCD, we have:
AD = BC (Already proved above)
AC = BD (Given in question)
DC = CD (Common side)
So, ΔADC ≅ ΔBCD (Side side side Congruence rule)
And ∠ADC = ∠BCD (By CPCT - Corresponding parts of congruent triangles )
Also, ∠ADC + ∠BCD = 180 degree (sum of Co-interior angles is 180 degree)
hence, ∠ADC + ∠ADC = 180 degree
2∠ADC = 180 degree
∠ADC = 90 degree
From this, we know that one of the interior angles of a quadrilateral (ABCD) is a right angle.
In conclusion to this, above we have obtained that ABCD is a parallelogram with AB = BC = CD = AD and one of its interior angles is equal to 90 degree (right angle).
Therefore, ABCD is a square.
Hence, proved.
(ii) Derive the area of the square using diagonals.
Ans: In a square ABCD, d is the diagonal.
Also, s represents the side of the square.
Therefore, acc. to Pythagoras theorem: square of hypotenuse = sum of squares of other two sides.
Here, the diagonal 'd' divides the square into 2 triangles.
So, the diagonal is the hypotenuse in this case.
Hence, by pythagoras theorem,
d2 = s2 + s2
d2 = 2s2
d = √2s; s = d/√2.
Now from this equation, we can find the area of the square.
we know that, Area = s²
Also, s2 = (d/√2)² = d²/2.
So, the area of the square 'in terms of the diagonal' will be d²/2.
(Where d = length of the diagonal)
