The plane ax + by + cz = d, meets the coordinate axes at the points, A, B and C respectively. Area of triangle ABC is equal to :

$\frac{d^2 \sqrt{a^2+b^2+c^2}}{2|a b c|}$

$A=\left(\frac{d}{a}, 0,0\right), B=\left(0, \frac{d}{a}, 0\right), C=\left(0,0, \frac{d}{a}\right)$

Area of triangle OAB = $\Delta_1=\frac{1}{2} \frac{d^2}{|ab|}$

Area of triangle OBC = $\Delta_2=\frac{1}{2} \frac{d^2}{|bc|}$

Area of triangle OAC = $\Delta_3=\frac{1}{2} \frac{d^2}{|ac|}$

If area of triangle ABC be $\Delta$, then

$\Delta^2=\Delta_1^2+\Delta_2^2+\Delta_3^2=\frac{d^4}{4}\left(\frac{a^2+b^2+c^2}{a^2 b^2 c^2}\right)$

$\Rightarrow \Delta=\frac{d^2 \sqrt{a^2+b^2+c^2}}{2|a b c|}$