The
Area of a Square
represents the total space it covers within its four equal sides. This concept is not just a part of geometry but a practical tool we encounter in daily life.
Whether it's figuring out how many tiles you need for your floor or determining the size of a garden plot, the area helps us measure and utilize space efficiently.
By learning to calculate the area, we can solve everyday problems effortlessly. In this article, we will explore the concept of the area of a square, including practical examples, steps to calculate the area of a square using side and diagonal formulas, and its units.
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Square in Math
A square is a two-dimensional shape that belongs to the family of quadrilaterals. It is defined by four sides of equal length, with each interior angle measuring 90°.
The opposite sides of the square are parallel to each other. One of its unique properties lies in its diagonals, which are equal in length and intersect at right angles.
This symmetrical and balanced shape is found in many objects that we use in everyday life, such as tiles, photo frames, chessboards, tabletops, and more.
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Properties of Square
Here are some common properties of a square:
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A square has four sides and four vertices, making it a type of quadrilateral.
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All sides are of equal length, and opposite sides are parallel.
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Each interior angle measures 90°, and the sum of all interior angles is 360°.
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The diagonals are equal in length, bisect each other at 90°, and divide the square into two congruent triangles.
What is the Area of a Square?
The area of a square refers to the amount of flat space enclosed within its four equal sides. In other words, it helps measure how much surface a square-shaped object occupies.
For example, think of a square garden.
The area of the garden represents the total space available within its boundaries for planting grass, flowers, or other decorations. The unit of measurement for area is always in square units, such as square meters, square centimeters, or square feet.
Area of a Square Formula
The area of a square formula helps calculate the space enclosed within its four equal sides. It is expressed as:
Area = Side × Side (square units)
If the diagonal of the square is given, the area of a square formula becomes:
Area = (Diagonal²) ÷ 2 (square units)
Example 1: Using the Side
Let’s consider a square garden with each side measuring 8 feet. Using the formula:
Area = Side × Side = 8 × 8 = 64 square feet.
The garden occupies a total area of 64 square feet.
Example 2: Using the Diagonal
For a square photo frame with a diagonal of 10 inches:
Area = (Diagonal²) ÷ 2 = (10 × 10) ÷ 2 = 100 ÷ 2 = 50 square inches.
Area of Square Formula Using Diagonal Derivation
The formula for finding the area of a square using its diagonal can be derived using the Pythagoras theorem.
Let the side of the square be "a" units and the diagonal be "d" units. The diagonal divides the square into two equal right-angled triangles.
In each triangle, the diagonal is the hypotenuse, and the two sides are the base and perpendicular.
According to the Pythagoras theorem:
Hypotenuse² = Base² + Perpendicular²
Substituting the values
:
d² = a² + a²
Simplify:
d² = 2a²
For example, if the diagonal is 10 units, then:
a² = d² / 2 = 10² / 2 = 100 / 2 = 50
The area of the square is given by a², so:
Area = 50 square units
Thus, the formula to calculate the area of a square using its diagonal is:
Area = (Diagonal²) ÷ 2 square units.
Area of a Square Solved Examples
Example 1: Find the area of a square if the length of the diagonal is 14 cm.
Solution:
Given:
Diagonal length (d) = 14 cm
The formula for the area using the diagonal is:
Area = (1/2) × d
2
Area = (1/2) × 14
2
Area = (1/2) × 196
Area = 98 cm²
The area of the square with a diagonal of 14 cm is 98 cm².
Example 2: Find the area of a square if the perimeter is 40 cm.
Solution:
Given:
Perimeter = 40 cm
The side length of the square can be found using the formula:
Side = Perimeter ÷ 4
Side = 40 ÷ 4 = 10 cm
Now, calculate the area using the formula:
Area = Side²
Area = 10² = 100 cm
The area of the square with a perimeter of 40 cm is 100 cm².
Example 3: A square playground costs $15 per square meter to lay grass. If each side of the playground measures 25 m, find the total cost.
Solution:
Given:
Side length (s) = 25 m
Cost per square meter = $15
First, find the area of the playground:
Area = Side²
Area = 25 × 25 = 625m
Now, calculate the total cost:
Total Cost = Area × Cost per square meter
Total Cost = 625 × 15 = $9375
The total cost to lay grass in the square playground is $9375.
Example 4: How many square tiles of 5 m side are required to cover a square hall of 50 m side?
Solution:
Given:
Side of the hall = 50 m
Side of one tile = 5 m
Find the area of the hall:
Area of hall = Side² = 50 × 50 = 2500m²
Find the area of one tile:
Area of one tile = Side² = 5 × 5 = 25 m²
Now, calculate the number of tiles needed:
Number of tiles = (Area of hall) ÷ (Area of one tile)
Number of tiles = 2500 ÷ 25 = 100
The total number of tiles required is 100.
Example 5: Calculate the area of a square if the perimeter is 60 cm and find the cost to paint it at $3 per square centimeter.
Solution:
Given:
Perimeter = 60 cm
Cost of painting per square cm = $3
Find the side length:
Side = Perimeter ÷ 4 = 60 ÷ 4 = 15 cm
Find the area:
Area = Side² = 15×15 = 225 cm
²
Calculate the total cost:
Total Cost = Area × Cost per square cm
Total Cost = 225 × 3 = $675
The cost to paint the square is $675.
