# Angle Between Two Planes Calculator

  

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.

 Plane (1):    $a_1$ = 1 , $b_1$ = 2 , $c_1$ = 3 Plane (1):    $a_2$ = -2 , $b_2$ = 3 , $c_2$ = 3 Number of Decimals = (4

## Outputs

More
Online Geometry Calculators and Solvers .