What is a Square?
Definition: A square is a two-dimensional shape which has four sides of equal length. The opposite sides of a square are parallel to each other and all four interior angles are right angles.
Paper napkins, chess boards, cheese slices, floor tiles, and so on are some real-life examples of square shaped objects.
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Properties of a Square
So now that we know the definition of a square, let’s look at some of its properties.
- A square is a quadrilateral with four sides and four vertices.
- All four sides of a square are equal in length.
- The opposite sides of a square are parallel to each other.
- All the diagonals of a square have the same length.
- In a square, the diagonals are longer than the sides.
- Each vertex of a square has a 90° internal angle.
- The sum of all interior angles of a square is 360°.
- The diagonals of a square intersect each other at 90°.
- A square can be divided into two equal triangles by the diagonals.
And now let’s move on to the next section to learn some basic formulas related to squares.
Read more perimeter of a square
Basic Formulas Related to Squares
Area of a Square:
The area of a square is defined as the measure of space occupied by the square. This area can be determined by the equation,
Area of a square = a\(^2\)
Where ‘a’ is the measure of the side of a square. Additionally, area is measured in square units.
Perimeter of a Square:
The perimeter of a square is the total length of the boundary of a square. It can be calculated by adding up the length of all the four sides.
Perimeter of a square = a + a + a + a
= 4a
Perimeter is measured in the same units as the unit of side length of the square.
Diagonal of a Square:
A line segment that connects two opposite vertices of a square is known as its diagonal. The formula for the length of the diagonal of square is:
Diagonal of a square (d) = \(\sqrt{2}\) × a
Rapid Recall
Solved Examples on Square
Example 1: A piece of land is in the shape of a square with side length 200 meters. What will be the cost of fencing the land at the rate of 10 cents per meter?
Solution:
The total cost of fencing can be determined by multiplying the boundary length of the land by the cost of fencing per meter.
Total boundary length is the perimeter of the land.
Perimeter of the square land = 4a [Formula for perimeter of square]
= 4 x 200 m [Substitute 200m for a]
= 800 m
So, the total cost of fencing = 800 x Cost of fencing per meter
= 800 x 10
= 8000
Therefore, the total cost of fencing the crop land is 8000 cents or $80.
Example 2: Lisa and Matt are playing chess on a wooden chessboard whose side equals 15 cm. Can you find the perimeter of the chessboard?
Solution:
We know that the shape of a chessboard is square.
As stated, the side of the chess board is 15 cm. So, each side will be of the same length.
Then, we can directly apply the perimeter of a square formula to get the answer.
That is,
Perimeter of a square = 4a
= 4 x 15 [Substitute the value of each side]
= 60
Hence, the perimeter of the chess board is 60 cm.
Example 3: Calculate the length of the diagonal of a square whose side equals 20 centimeters.
Solution:
The given square has a side length of 20 cm.
The formula to find the length of a diagonal is
Diagonal of Square (d) = \(\sqrt{2}\) × a
= \(\sqrt{2}\) × 20 [Substitute 20 for a]
= 1.414 × 20 [Substitute 1.414 for \(\sqrt{2}\) ]
= 28.28
Hence, the length of the diagonal is 28.28 cm.
A quadrilateral is a type of polygon with 4 sides, 4 vertices and 4 interior angles.
Yes, a square is a polygon.
A square, like any other quadrilateral, has 2 diagonals.
The difference between square and rectangle is that all the sides in a square are equal. However, in a rectangle only the opposite sides are equal.
No. Area is the measure of space occupied by the square. The perimeter of a square refers to the total length of its outer boundary.
Yes, all the four angles of a square are right angles.