Question

How to find the height of a triangle without area?

Answer 1

Calculate the height of a triangle without area

Assuming all three sides are given. Drop a perpendicular $BO$ on side $AC$.

In $△\mathrm{ABC}$:

$\angle \mathrm{A}={\cos }^{-1}\left(\frac{{\mathrm{b}}^2+{\mathrm{c}}^2-{\mathrm{a}}^2}{2\mathrm{bc}}\right)\left[\because \mathbf{cos}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\frac{{\mathbf{b}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{c}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathbf{a}}^{\mathbf{2}}}{\mathbf{2}\mathbf{b}\mathbf{c}}\mathbf{}(\mathrm{Cosine}\mathrm{Rule})\right]$

Now, In $△\mathrm{AOB}$ : $\sin \left(\mathrm{A}\right)=\frac{\mathrm{h}}{\mathrm{c}}...\left(\mathrm{i}\right)$

Put value of $\angle \mathrm{A}$ in $\left(\mathrm{i}\right)$

$\Rightarrow \mathrm{h}=\mathrm{c}\times \sin \left({\cos }^{-1}\left(\frac{{\mathrm{b}}^2+{\mathrm{c}}^2-{\mathrm{a}}^2}{2\mathrm{bc}}\right)\right)$

Hence, using cosine rule height can be calculated without area of triangle.