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 $.