Square Formula: Area, Perimeter, Properties & Solved Examples

Square is a quadrilateral that has all four sides equal along with equal angles according to Euclidean Geometry. A square is derived from a rectangle and hence can also be defined as a rectangle having two equal adjacent sides. Square is one of the basic quadrilaterals having a definite structure and hence a quadrilateral can be considered to be a square if it poses the properties of a square such as all sides being equal and all angles being equal. A square can be defined as a quadrilateral with all four angles being right angles, i.e, 90 degrees. Given below is the detailed information on the square formulas such as area, perimeter, length of the diagonals along with solved examples.

Key Terms: Square Formula, Area of Square, Diagonal of Square, Properties of Square, Characteristics of Square

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Square Formula

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Square is indeed one of the most recurring and basic geometrical shapes. Given below is a list of various formulae of a square that can be used in day-to-day calculations, asked in question papers, and hence are the building blocks of Geometry.

Square

For better understanding, we will take the side of a square as ‘a’ unit.

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Area of a Square

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The area of a square is the product of the length of two sides of a square and hence can be mathematically represented as

• Area of a Square = side x side = (a²)

Area of a Square

For example, Calculate the area of a square with a side of 20cm.

As we know the area of the square is the square of the side of a square hence,

Area = 20²

Area = 400 cm²

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Perimeter of a Square

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The perimeter of a polygon can be defined as the sum of all sides of a polygon. Hence, the perimeter of a square is calculated as

• Perimeter of a square = a+a+a+a or P= 4a

For example, A square has a side of 10 cm. What will be its perimeter?

Here, a = 10 cm

Perimeter (P) = a+a+a+a = 4a

Therefore, P = 4*10

Perimeter (P) =40 units

Also Read: Difference Between Area and Perimeter

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Diagonals of a Square

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A diagonal is a straight line joining two opposite corners of a straight-sided shape polygon.

Diagonal is calculated by using the formula:

• Diagonal (D) = a√2 units

Diagonals of a Square

For example, A square has a side of 50 cm, find its diagonal.

Here, a = 50cm

As we know Diagonal (D) = a√2 units,

Therefore, D = 50√2 cm

• Derivation of Square Diagonal Formula

Here is the derivation for the diagonal formula of the square,

From the above diagram, we can see that a square is divided into two equal right-angled triangles. Right angle triangle because every side makes an angle of 900 with the other side giving the formation of a right-angle triangle. And because it is a square, all four sides of the rectangle will be equal i.e., AB=BC=CD=AD=a.

Applying Pythagoras theorem in Triangle ABC,

AC² = AB² + BC²

As AB=BC=CD=AD=a.

Therefore, AC² = AB² + AB² -------------- (AB=BC)

AC² = a² + a² -------------- (AB=a)

AC² = 2a²

Taking square root on both sides

AC = a√2

Also Read: Area of Square Using Diagonal

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Properties of a Square

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Square is a unique case of a rhombus, a kite, a trapezoid, a parallelogram, a quadrilateral, and a rectangle, and thus has the following of properties:

• Diagonal of a square cut each other in half and meet at a 90° angle.
• A square's diagonals cut over its angles.
• The opposite sides of a square are parallel as well as equal in length.
• All four angles of a square are equal (i.e., 900)
• Sides of a square are equal in terms of length.
• Diagonals of squares are equal.

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Characteristics of a Square

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If and only if a convex quadrilateral is one of the following, it is a square:

• Two adjacent equal sides form a rectangle.
• A rhombus with a right-angled vertex
• A rhombus having identical angles on all sides
• A parallelogram with two adjacent equal sides and one right angle.
• Four equal sides and four right angles make up a quadrilateral.
• Diagonals are equal and perpendicular bisectors of each other.

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Things to Remember

• Area of a square (A) = (a²)
• Perimeter (P) = 4a
• Diagonal (D) = a√2 units.
• A square's diagonals are √2 (approximately 1.414) times the length of one of the square's sides. The square root of 2 (also known as Pythagoras' constant) was the first number to be demonstrated to be irrational.
• A parallelogram with equal diagonals that bisect the angles is also known as a square.
• A square is a figure that is both a rectangle (right angles) and a rhombus (equal edge lengths).