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# Rectangle|Definition & Meaning

**Definition **

A **rectangle** is a flat (two dimensional) shape with **four** straight sides such that the length of the **opposite** sides are equal in measure and it possesses right angles for all of its interior angles. Therefore, both pairs of opposite sides are **parallel** and each side is **perpendicular** to all adjacent sides.

## What Is a Rectangle?

A **polygon** having four sides or 4 **corners** are called a **rectangle.** If the edge-to-edge distance of all the sides of a polygon is equal in length, then the formed shape is known as a **square.** Whereas a **rectangle** has two pairs of **equal-length** sides, and the **equal sides** are **opposite** to each other, which differentiates it from a **square.**

A **rectangle** is a **two-dimensional** shape, which means it can not be transformed into a **3D surface.** In **3** **dimensional geometry,** a rectangle is sometimes referred to as a **cube.** All the inner angles of a **rectangle** are **equal** in **measure** of 90$^{\circ}$. Since it is constructed in two **dimensions,** it is illustrated by two dimensions, **length(L)** and **width(W).**

**Diagonals of a Rectangle **

The line segment that begins from one vertex and ends at the opposite non-adjacent vertex is known as a **diagonal**. A rectangle has **two** diagonals that bisect each other at their point of **intersection**. Both diagonals are **congruent** and equal in length. They are not mutually perpendicular. A diagonal divides the rectangle into two **right-angled** triangles.

**Finding the Length of a Diagonal**

Consider the right angle triangle ABC, diagonal BC is the **hypotenuse**, AB (length) is the **base,** and AC (width) is the **perpendicular** of the triangle.

Let d represent the diagonal, l represents the length, and w represent the width. Then, by using **Pythagoras’** **theorem**, we have

Diagonal^{2} = length^{2} + width^{2}

d^{2} = l^{2} + w^{2}

d = $\sqrt{l^2 + w^2}$

Let’s take a look at how to find the length of diagonals of a rectangle having two sides of 10cm in length and the other two of 6cm in length. By using the formula,

d = $\sqrt{l^2 + w^2}$

= $\sqrt{10^2 + 6^2}$

= $\sqrt{100 + 36}$

= $\sqrt{136}$

**d = 11.6 cm**

Therefore, the length of both diagonals is 11.6cm.

**Properties of Rectangle **

Following are the properties of a rectangle

- A rectangle is a
**quadrilateral**. - It has 4 sides, 4
**angles**, and 4**vertices**. **Opposite**sides of a rectangle are equal.- Diagonals of a rectangle
**bisect**each other. - The
**length**of the diagonals is equal. - Each angle of the rectangle is
**90 degrees**. - If you sum up all the
**interior**angles, the result would be 360**degrees.** - The
**longer**side of the rectangle is its length. - The
**shorter**side of the rectangle is its width.

**3D Rectangle**

A 3-dimensional rectangle is called a rectangular **prism** and **cuboid**. It has 6 **faces**, 12 **edges**, and 8 **vertices**.

**Area of Rectangle **

**The area** of the rectangle is the entire region surrounded by the **rectangle’s** **boundaries** in a 2-dimensional plane. It is the product of length and width. The area occupied by **rectangular-shaped** objects such as a television, blackboard, door, refrigerator, etc., can easily be calculated by utilizing the formula for the area of a rectangle:

**Area of rectangle = length * width**

Consider an example of a television whose **length** is 42 inches and **width** is 20 inches. To figure out the exact area of a rectangle, we use the following formula,

**Area** = length * width

= 42 * 20

= 840 inch^{2}

**The Perimeter of a Rectangle**

The perimeter of a rectangle is the path or **distance** covered by the **boundary** of the rectangle in a 2-dimensional plane. If two adjacent sides of a **rectangle** are summed up, we get the **perimeter** of a rectangle, which can be found using the mathematical **formula** given below,

**Perimeter** of rectangle = length + length + width + width

**= 2*(length) + 2*(width)**

**= 2(length + width)**

Consider an example of a gardener who wants to find out how much wire would be required to put a **fence** around his garden. The **length** of the garden is 15 feet, and its **width** is 6 feet.

The answer to this problem can be found by calculating the **perimeter** of the garden as follows:

Length = 15 feet

Width = 6 feet

Perimeter = 2(length + width)

**= 2(15 + 6)**

**= 2(21)**

**= 42 feet**

Hence, 42 feet of wire would be required to put a fence around the garden.

**Quadrilateral**

A **quadrilateral** is a four-sided plane closed figure. It has four angles and four vertices. It is a (four-sided) **polygon** also known as a **tetragon**. The total measurement of all the interior angles comes out to be 360 degrees.

A quadrilateral is a combination of two Latin words, where **Quadri **stands for four, and **lateral** is described as sides. Hence, the name itself gives the description of this plane figure.

A **rectangle** is a type of quadrilateral. Some other **types** of quadrilaterals are square, parallelogram, rhombus, trapezium, and kite.

**Square**

A Square is a **quadrilateral,** but it differs from a rectangle as all of its sides are **equal** in length. Like a rectangle, a square has **four** equal angles of 90 degrees each. **Diagonals** of a square cross each other at a common point that has a 90-degree angle, thus bisecting each other **perpendicularly**.

**Parallelogram**** **

A parallelogram is a **quadrilateral**. It is like a rectangle, but its diagonals are **unequal**. Its opposite sides are **parallel** and equal. Diagonals bisect each other at the **midpoint** and shape two **congruent** triangles. A parallelogram has **opposite** equal angles.

We can say that a rectangle possesses all the mathematical properties of a parallelogram, but a **parallelogram** does not have all the properties of a rectangle. So a rectangle is a parallelogram, but a parallelogram may not be a **rectangle**.

** Rhombus **

A rhombus is a **quadrilateral** having all four sides equal, **opposite** sides parallel, and opposite **angles** equal. **Diagonals** of a rhombus cross each other at a common point that has a 90-degree angle, thus bisecting each other **perpendicularly**. All **squares** are rhombus, but the converse may not be true.

**Trapezium **

A trapezium is a **combination** of parallel and non-parallel sides. A quadrilateral having only one pair of **non-adjacent** **parallel** sides and having the other two as **non-parallel** sides is a trapezium. The sides that are parallel to each other are known as **bases,** whereas the other two non-parallel sides are known as **legs.** A parallelogram is a trapezium but a **trapezium** is not a parallelogram.

**Kite **

A kite is a four-sided figure, just like a tilted parallelogram but with two pairs of sides having **equal lengths**. If we draw the diagonals of a kite, they will behave just like a normal rectangular diagonal by **intersecting** each other at right angles.

The interior opposite angles of a kite are **obtuse** and equal in measure.

**Solved Examples Involving Rectangular Shapes**

### Example 1

Find the length of a rectangle whose area is 400 feet^{2} and whose width is 16 feet.

### Solution

Given:

Area = 400 feet^{2}

Width = 16 feet

Length = ?

As we know that:

area = length * width

Putting the values of area and length in the above formula, we get:

**400 = length * 16**

**400/16 = length**

**25 feet = length **

**Length = 25 feet**

Therefore, the rectangle has an area of 400 feet and a width of 16 feet, has a length of 25 feet.

### Example 2

A rectangular prism has a length of 20 inches and a width of 15 inches. Find the area and perimeter of the prism.

### Solution

Given:

Length = 20 inches

Width = 15 inches.

Area =? ; Perimeter =?

For area, we have the formula:

Area = Length * Width

**= 20 * 15**

**= 300 inches**^{2}

Now for perimeter, the formula is:

Perimeter = 2(length + width)

**= 2(20 + 15)**

**= 2(35)**

**= 70 inches **

Therefore, the rectangular prism has an area of 300 inches^{2} and a perimeter of 70 inches.

*All images were created with GeoGebra.*