# Properties of Rectangle: A Complete Guide

Understanding the properties of rectangles unveils essential principles in geometry, where this quadrilateral can help you understand basic ideas and formulas. Defined by its four right angles and parallel sides of equal length, a rectangle is symmetry and efficiency. Terms such as length, width, diagonal, and perimeter are very important in describing its dimensions and characteristics. These properties not only support foundational geometric concepts but also find practical application in fields ranging from architecture and engineering to art and design. This comprehensive guide explores the basic details of properties of rectangles, encompassing their mathematical formulas, spatial relationships, and diverse applications, offering a full picture of this fundamental geometric shape.

## Definition of Rectangle

A rectangle is a planar shape that is two-dimensional and has four sides for each side. One definition of a rectangle is a polygon with four sides, in which the opposing sides are parallel to one another and equal to one another. It belongs to the category of quadrilaterals in which each of the four angles is either perpendicular to the other or equal to ninety degrees. As a specific kind of parallelogram, the rectangle is characterized by having all of its angles being equal. A square is a rectangle that has four sides that are also equal to one another.

A rectangle is a four-sided polygon, or quadrilateral, with the following properties of Rectangle:

1. Four Right Angles: Each of the four interior angles is 90 degrees.
2. Opposite Sides are Equal and Parallel: The length of each pair of opposite sides is the same, and these sides are parallel to each other.
3. Diagonals are Equal: The diagonals (the lines connecting opposite corners) of a rectangle are equal in length.
4. Perimeter: The perimeter of a rectangle is the total distance around the shape, calculated as 2×(length+width).
5. Area: The area of a rectangle is the amount of space it covers, found by multiplying the length by the width (Area=length×width).

## Properties of Rectangle with Formulas

A rectangle is a fundamental geometric shape characterized by its four right angles and parallel sides of equal length. Understanding its properties is crucial in various fields such as mathematics, architecture, and engineering. Below is a table summarizing the key properties of rectangle.

## Properties of Rectangle Solved Examples

Here are 10 solved examples based on properties of rectangles, covering various aspects such as area, perimeter, and properties.

Q1: Find the area of a rectangle with a length of 8 units and a width of 5 units.

Solution: Area = length × width

= 8 × 5 = 40 square units.

Q2: Find the perimeter of a rectangle with a length of 12 units and a width of 7 units.

Solution: Perimeter = 2(length + width)

= 2(12 + 7)

= 2 × 19 = 38 units.

Q3: A rectangle has a length of 15 units and an area of 105 square units. Find the width.

Solution: Area = length × width

therefore, width = Area / length

= 105 / 15 = 7 units.

Q4: A rectangle has a width of 9 units and a perimeter of 52 units. Find the length.

Solution: Perimeter = 2(length + width)

therefore, 52 = 2(length + 9).

Solving for length: 52 / 2 = length + 9

26 = length + 9;

length = 26 – 9 = 17 units.

Q5: Find the length of the diagonal of a rectangle with a length of 6 units and a width of 8 units.

Solution: Diagonal = √(length² + width²)

= √(6² + 8²)

= √(36 + 64) = √100 = 10 units.

Q6: A rectangular field has a length of 20 meters and a width of 15 meters. If the cost of fencing is \$5 per meter, find the total cost.

Solution: Perimeter = 2(length + width)

= 2(20 + 15)

= 2 × 35 = 70 meters.

Total cost = 70 × 5 = \$350.

Q7: A rectangle has a perimeter of 48 units and a length of 14 units. Find the area.

Solution: Perimeter = 2(length + width)

therefore, 48 = 2(14 + width).

Solving for width: 48 / 2 = 14 + width

24 = 14 + width;

width = 24 – 14 = 10 units.

Area = length × width

= 14 × 10 = 140 square units.

Q8: A rectangle has a length of 5 units and a width of 3 units. If both dimensions are increased by 2 units, find the new area.

Solution: New length = 5 + 2 = 7 units

New width = 3 + 2 = 5 units.

New area = 7 × 5 = 35 square units.

Q9: The length and width of a rectangle are in the ratio 4:3. If the perimeter is 56 units, find the dimensions.

Solution: Let length = 4x and width = 3x.

Perimeter = 2(length + width)

= 2(4x + 3x) = 2(7x) = 14x.

Solving for x: 14x = 56

x = 56 / 14 = 4.

Length = 4x = 4(4) = 16 units

Width = 3x = 3(4) = 12 units.

Q10: A rectangle has an area of 80 square units and a length of 10 units. Find the perimeter.

Solution: Area = length × width

therefore, width = Area / length

= 80 / 10 = 8 units.

Perimeter = 2(length + width)

= 2(10 + 8)

= 2 × 18 = 36 units.

Also Read: Formula of Profit and Loss: Percentages

## FAQs

What is the formula for Area of rectangle?

Area of a rectangle = length × width .

How to prove a rectangle?

Here are three simple ways to do it:
1. Prove that all the angles are 90°.
2. Prove that two opposite angles are 90° and that one pair of sides is parallel.
3. Prove that the diagonals cut each other in half and are all the same length.

What is the principle of rectangle?

Both sides of a rectangle are parallel and equal. Since a rectangle is like a parallelogram in every way, it is also called a parallelogram. The number of each of the rectangle’s four inside angles is 90°, so they are all the same. All of a rectangle’s inside sides add up to 360°.

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