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We multiply by 5 because the absolute value of the determinant of the transformation matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$ is 5. The determinant represents the area scaling factor of the linear transformation.

Conditions: The transformation matrix is $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.

We multiply by 5 because the absolute value of the determinant of the transformation matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$ is 5. The determinant represents the area scaling factor of the linear transformation.

Conditions: The transformation matrix is $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.