A free online calculator, showing all steps, to calculate the angle \( \alpha \) between two planes is presented.

Formulas Used in the Calculator

Plane (1) and plane (2) have the following equations \( \quad a_1 x + b_1 y + c_1 + d_1 = 0\) and \( \quad a_2 x + b_2 y + c_2 + d_2 = 0\) respectively.
The vectors \( \vec {n_1} \) and \( \vec {n_2} \) normal to the planes (1) and (2) defined by their equation above are given by their components as:
\( \vec {n_1} \; = \; \lt a_1 , b_1 , c_1 \gt \)
\( \vec {n_2} \; = \; \lt a_2 , b_2 , c_2 \gt \)
Angle \( \alpha \) between the two planes is equal to the angle between vectors \( \vec {n_1} \) and \( \vec {n_2} \) and its cosine is given by
\[ \large \color{red} {\cos \alpha = \dfrac{ \vec {n_1} \cdot \vec {n_2} }{| \vec {n_1} | \cdot | \vec {n_2} | } = \dfrac{a_1 \cdot a_2 + b_1 \cdot b_1 + c_1 \cdot c_2 }{| \vec {n_1} | \cdot | \vec {n_2} | } } \]
The magnitudes \( | \vec {n_1} | \) and \( | \vec {n_2} | \) are given by
\( | \vec {n_1} | = \sqrt {a_1^2 + b_1^2 + c_1^2 } \)
\( | \vec {n_2} | = \sqrt {a_2^2 + b_2^2 + c_2^2 } \)
Use the inverse cosine function to express angle \( \alpha \) made by the two vectors as
\[ \large \color{red} {\alpha = \arccos \left (\dfrac{ a_1 \cdot a_2 + b_1 \cdot b_1 + c_1 \cdot c_2 }{\sqrt {a_1^2 + b_1^2 + c_1^2 } \cdot \sqrt {a_2^2 + b_2^2 + c_2^2} } \right) } \]

Use of the Calculator

Enter the coefficients \( a_1 \), \( b_1 \) and \( c_1 \) of plane (1) and coefficients \( a_2 \), \( b_2 \) and \( c_2 \) of plane (2) and press "Calculate". The outputs are the magnitudes \( | \vec {n_1} | \) and \( | \vec {n_2} | \), the dot product \( \vec {n_1} \cdot \vec {n_2} \) and angle \( \alpha \). You may also enter the number of decimal places required.