This **article aims** to find the **area of the parallelogram**. This article uses the concept of the **area of the parallelogram**. **A parallelogram** **bounds a parallelogram**‘s area in a given **two-dimensional space**. As a reminder, a parallelogram is a particular type of quadrilateral with four sides, and the pairs of opposite sides are parallel. In** parallelogram**, opposite sides have the same **length**, and** opposite angles** have equal measures. Since a rectangle and a parallelogram have similar properties, the **area of the rectangle** is equal to the area of a **parallelogram**.

To find **area of a parallelogram,** multiply the** **perpendicular base by its **height. **It should be noted that the base and altitude of a parallelogram are **perpendicular** to each other, while the lateral side of a **parallelogram is not perpendicular to the base**.

\[ Area = b \times h \]

Where $ b $ is the **base** and $ h $ is the **height of the parallelogram. **

**Expert Answer**

A **parallelogram** can be described by $ 4 $ **vertices** or $ 2 $ **vectors**. Since we have $ 4 $ vertices $ (ABCD) $, we find the **vectors** $ u $, $ v $ that describe the **parallelogram.**

\[ A = ( 0 , 0 ) \]

\[ B = ( 5 , 2 ) \]

\[ C = ( 6 , 4 ) \]

\[ D = ( 11 , 6 ) \]

\[ u = AB = \begin{bmatrix}

5 \\

2

\end{bmatrix} \]

\[ v = AC = \begin{bmatrix}

6 \\

4

\end{bmatrix} \]

**Area of parallelogram** is the absolute value of the **determinant.**

\[ \begin{bmatrix}

u _ { 1 } & v _ { 1 } \\

u _ { 2 } & v _ { 2 }

\end{bmatrix} = det \begin{bmatrix}

5 & 6 \\

2 & 4

\end{bmatrix}= 20 \: – \: 12 = 8 \]

The **area of the parallelogram** is $ 8 $.

**Numerical Result**

The **area of the parallelogram** is $ 8 $.

**Example**

**Find area of the parallelogram whose vertices are given. $ ( 0 , 0 ) $, $ ( 5 , 2 ) $, $ ( 6 , 4 ) $ , $ ( 11 , 6 ) $**

**Solution**

A **parallelogram** can be described by $ 4 $ **vertices** or $ 2 $ **vectors**. Since we have $ 4 $ vertices $ ( ABCD ) $, we find the **vectors** $ u $, $ v $ that describe the **parallelogram.**

\[ A = ( 0 , 0 ) \]

\[ B = ( 6 , 8 ) \]

\[ C = ( 5 , 4 ) \]

\[D = ( 11 , 6 ) \]

\[ u = AB = \begin{bmatrix}

6\\

8

\end{bmatrix} \]

\[ v = AC = \begin{bmatrix}

5\\

4

\end{bmatrix} \]

**Area of parallelogram** is the absolute value of the **determinant.**

\[ \begin{bmatrix}

u _ { 1 } & v _ { 1 } \\

u _ { 2 } & v _ { 2 }

\end{bmatrix} = det \begin{bmatrix}

6 & 5 \\

8 & 4

\end{bmatrix}= 24 \: – \: 40 = 16 \]

The **area of the parallelogram** is $ 16 $.