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# Rectangle

When you look at an object, some questions come to mind and right there you give quick answers to a number of them. One of such questions answered is the shape of the object.

In this article, we will explore the definition of a rectangle, its properties, the formulas for the perimeter and area of a rectangle, and examples of their application.

## Definition of rectangle

A rectangle is a quadrilateral with four sides and four angles, where all the interior angles are right angles (90 degrees).

A rectangle is a special case of a parallelogram. In other words, what makes a parallelogram become a rectangle is having its sides perpendicular to each other. This can be illustrated in the image below.

Rectangle - StudySmarter Original

We can notice that the opposite sides AB and CD are equal in size and parallel, the same for BC and AD. Moreover, the four sides are perpendicular to each other, thus, the quadrilateral is a rectangle.

## Rectangles Properties

A rectangle being a parallelogram has all the properties of a parallelogram, but being a special case of it, it has its own unique properties that make it the geometric shape it is.

To better understand the properties of a rectangle, let's consider the following rectangle ABCD in the image below.

Rectangle ABCD - StudySmarter Original

 Property Example 1. Opposite sides of a rectangle are equal and parallel. AB = CD, and AB is parallel to CD.Likewise, AD = BC, and AD is parallel to BC. 2. All four angles in a rectangle are right angles. $\angle A=\angle B=\angle C=\angle D=90°$ 3. The sum of all interior angles of a rectangle is 360º. $\angle A+\angle B+\angle C+\angle D=360°$ 4. The diagonals of a rectangle are equal in length and bisect each other – they intersect each other in their middle. AC and BD are the diagonals of rectangle ABCD.AC = BDAC bisects BD and BD bisects AC.

## Construction of a rectangle

For the construction of a rectangle, follow these steps.

Step 1: Draw a straight line (R), then place 2 points A and B on the line.

Step 2: Draw 2 perpendicular lines (S) and (T), passing by the two points A and B.

Step 3: Locate two points C and D respectively on the two lines (S) and (T). However, C and D must be on the same level.

The three steps mentioned earlier can be illustrated in the drawing below:

Image resulting from steps 1, 2 and 3 - StudySmarter Original

Step 4: Draw a straight line joining the two points C and D, as shown in the image below:

Image resulting from step 4 - StudySmarter Original

After reaching step 4, you will notice that the 4 points A, B, C, and D will form a rectangular shape.

## Formula for the area of a rectangle

The area of a flat shape or the surface of an object can be defined in geometry as the space occupied by it.

The area of a shape is usually measured considering the number of unit squares that cover the surface of the shape. Square centimeters, square feet, square inches, and other similar units are used to measure area.

Given a rectangle with height h and base b, its area will be equal to:

$A=b×h$.

Find the area of the rectangular shape in the image below. Consider the square composed of 25 smaller squares the square of side 1 unit.

We can notice that the height of the rectangle is equal to 2 unit squares, so its length is 2 units. Similarly, the base of the rectangle is 5 units. So the area of this rectangle can be calculated by multiplying the height by the base:

A= 2 unit × 5 unit = 10 unit2

## Formula for the perimeter of a rectangle

The perimeter of a shape is the distance around its outside.

Consequently, the shape's perimeter is calculated by summing the lengths of all its sides. The same concept also applies to a rectangular shape. So, the total length of all the sides of a rectangle is known as the perimeter.

A rectangle has its opposite sides equal to each other (one of its properties). Thus, the rectangle's perimeter of a rectangle with sides of lengths a, b, a, b is P = a + b + a + b, or P = 2a + 2b, or even P = 2 (a + b).

So, we just need to calculate the lengths of two sides to find the perimeter of a rectangle since opposite sides of a rectangle are always equal.

Find the perimeter and the area of the shape illustrated in the image below:

Step 1: Try to identify the rectangle shapes. We can notice that 2 rectangles are present in the shape above. The rectangles identified are illustrated in the image below:

The following properties are checked to make sure that the shapes identified are rectangles:

1. The opposite sides are equal and parallel. For example, the first rectangle identified has the opposite sides parallel and equal to 8, and the other two opposite sites also parallel and equal to 3.
2. All the angles are right angles, or in other words all the sides are perpendicular to each others.

The Perimeter of the first rectangle PA can be calculated as follows:

${P}_{A}=\left(4+4+3+3\right)cm=14cm$

The Perimeter of the second rectangle PB can be calculated as follows:

${P}_{B}=\left(10+10+5+5\right)cm=30cm$

The Perimeter of the overall shape PAB:

${P}_{AB}={P}_{A}+{P}_{B}=\left(14+30\right)cm=\mathbf{44}\mathbit{c}\mathbit{m}\mathbf{}\mathbf{}$

The Area of the first rectangle AA can be calculated as follows:

${A}_{A}=height×base=4cm×3cm=12c{m}^{2}$

The Area of the second rectangle AB can be calculated as follows:

${A}_{B}=height×base=5cm×10cm=50c{m}^{2}$

The Area of the overall shape:

${A}_{AB}={A}_{A}+{A}_{B}=12c{m}^{2}+50c{m}^{2}=\mathbf{62}\mathbit{c}{\mathbit{m}}^{\mathbf{2}}$

## Square and rectangle

You can notice in the figure below that a square and a rectangle are both quadrilateral with four sides.

A rectangle and a square - StudySmarter Original

A square and a rectangle have similar properties as illustrated in the table below:

 Properties Rectangle Square The four sides are equal X ✔ Opposite sides are equal ✔ ✔ Opposite sides are parallel ✔ ✔ Diagonals bisect each others ✔ ✔ Diagonals are perpendicular to each others X ✔ All angles are equal ✔ ✔ Opposite angles are equal ✔ ✔ Sum of two adjacent angles is 180 degrees ✔ ✔

### What Characterizes a Square as a Unique Rectangle?

As illustrated in the table above, a square is a special type of rectangle for the following reasons:

1. A square has all the properties of a rectangle.

2. The only two differences between the square and the rectangle are that a square has its diagonal perpendicular to each other, and all its sides are equal.

## Rectangle - Key takeaways

• A rectangle is also a quadrilateral with four sides and four angles.
• All four angles in a rectangle are right angles.
• Opposite sides in a rectangle are equal and parallel A rectangle's diagonals are equal and bisect each other's. They bisect each other means that they intersect each other in their middle.
• Consecutive angles in a rectangle are supplementary. Their summation is equal to 180 degrees.
• Given a rectangle with height equal to h and base equal to b, then its corresponding area will be equal to the multiplication of b by h.
• A rectangle has its opposite sides equal to each other. Thus, the specified rectangle's perimeter is 2(a + b).
• A square is a unique rectangle.

Any four-sided shape with all its interior angles being right angles are an example of rectangles.

To find the area of a rectangle, use the formula A = b × h, where b is the base and h is the height of the rectangle.

To find the perimeter of a rectangle, use the formula P = 2(a + b), where a and b are the lengths of the sides.

Yes, all squares are rectangles.

The formula for the area of the rectangle is A = b × h.

## Final Rectangle Quiz

Question

Can the diagonal of a rectangle be perpendicular?

No

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Question

In a rectangle the opposite sides are equal and parallel. Is is true or false?

True

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Question

In a rectangle at least 2 angles must be right angles. Is it true or false?

False

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Question

The perimeter of a flat shape or the surface of an object can be defined in geometry as the space occupied by it. Is is true or False?

False

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Question

All rectangles are squares. Is it true or false?

False

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