## Square – Introduction

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.

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## Definition of Square in Math

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

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## What is a Square in Math?

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.

## Properties of a Square

- It has 4 sides and 4 vertices.
- Its sides are equal in length.
- All interior angles are equal and right angles, which means that each angle measures 90°.
- The sum of all the interior angles is 360°.
- Its two diagonals bisect each other at right angles.

## Area and Perimeter of a Square

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.

**Area**

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.

**Perimeter**

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.

## Solved Examples

**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.

## Practice Problems

## Square

### If the side of a square chess board is 20 cm in length, find the board's perimeter.

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 $= 20$ cm.

Therefore, the perimeter $= 4 x 20 = 80$ cm.

### What is the measure of an interior angle of a square?

Since all the interior angles of a square are right angles, the measure of each one of them is 90 degrees.

### What is the area of a square with a side length of 25 cm?

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

It is given that the side $\text{s} = 25$ cm.

Therefore,

Area $= \text{s}^2 = 25 \times 25 = 625$ sq. cm.

## Frequently Asked Questions

**How to identify a polygon as a square?**

A polygon that is made of four equal sides, with all the interior angles measuring 90 degrees, is a square.

**Is the side of a square and its diagonal the same length?**

No, the side of a square and its diagonal aren’t of the same length. The diagonal of a square is greater in length than its side.

**If two squares have the same perimeter, will they have the same area?**

Yes, if two squares have the same perimeter, it means that they also have sides of the same length. This, in turn, implies that they also have the same area.