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Calculating the area of a parallelogram with only two dimensions given such its base, $b$ and diagonal lengths. Can this be done or we need another information?

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Yes, the area of a parallelogram can be determined using the diagonal lengths $d_1,d_2$ and the base length $b$.

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The expression for the area is $\frac12d_1d_2\sin A$, where $A$ is the angle between the diagonals. For calculating $\sin A$, first find $\cos A$ using the cosine formula,$$\cos A=\frac{\left(\frac{d_1}2\right)^2+\left(\frac{d_2}2\right)^2-b^2}{2\left(\frac{d_1}2\right)\left(\frac{d_1}2\right)}$$and then use the fact $\sin A=\sqrt{1-\cos^2A}$.

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