## Credits

Walchand College of Engineering (WCE), Sangli
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Vellore Institute of Technology (VIT), Bhopal
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## Angle between two planes Solution

STEP 0: Pre-Calculation Summary
Formula Used
angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2)))
∠C = acos(((l1*l2)+(m1*m2)+(n1*n2))/(sqrt((l1)^2+(m1)^2+(n1)^2)* sqrt((l2)^2+(m2)^2+(n2)^2)))
This formula uses 3 Functions, 6 Variables
Functions Used
cos - Trigonometric cosine function, cos(Angle)
acos - Inverse trigonometric cosine function, acos(Number)
sqrt - Squre root function, sqrt(Number)
Variables Used
Direction cosine 1 with respect to x axis - Direction cosine 1 with respect to x axis is the cosine of angle made by a line w.r.to x axis. (Measured in Hundred)
Direction cosine 2 with respect to x axis - Direction cosine 2 with respect to x axis is the cosine of angle made by a line w.r.to x axis. (Measured in Hundred)
Direction cosine 1 with respect to y axis - Direction cosine 1 with respect to y axis is the cosine of angle made by a line w.r.to y axis. (Measured in Hundred)
Direction cosine 2 with respect to y axis - Direction cosine 2 with respect to y axis is the cosine of angle made by a line w.r.to y axis. (Measured in Hundred)
Direction cosine 1 with respect to z axis - Direction cosine 1 with respect to z axis is the cosine of angle made by a line w.r.to z axis. (Measured in Hundred)
Direction cosine 2 with respect to z axis - Direction cosine 2 with respect to z axis is the cosine of angle made by a line w.r.to z axis. (Measured in Hundred)
STEP 1: Convert Input(s) to Base Unit
Direction cosine 1 with respect to x axis: 0.7 Hundred --> 0.7 Hundred No Conversion Required
Direction cosine 2 with respect to x axis: 0.7 Hundred --> 0.7 Hundred No Conversion Required
Direction cosine 1 with respect to y axis: 0.8 Hundred --> 0.8 Hundred No Conversion Required
Direction cosine 2 with respect to y axis: 0.6 Hundred --> 0.6 Hundred No Conversion Required
Direction cosine 1 with respect to z axis: 0.6 Hundred --> 0.6 Hundred No Conversion Required
Direction cosine 2 with respect to z axis: 0.6 Hundred --> 0.6 Hundred No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
∠C = acos(((l1*l2)+(m1*m2)+(n1*n2))/(sqrt((l1)^2+(m1)^2+(n1)^2)* sqrt((l2)^2+(m2)^2+(n2)^2))) --> acos(((0.7*0.7)+(0.8*0.6)+(0.6*0.6))/(sqrt((0.7)^2+(0.8)^2+(0.6)^2)* sqrt((0.7)^2+(0.6)^2+(0.6)^2)))
Evaluating ... ...
∠C = 0.137761607411346
STEP 3: Convert Result to Output's Unit
0.137761607411346 Radian -->7.89315868360975 Degree (Check conversion here)
7.89315868360975 Degree <-- Angle C
(Calculation completed in 00.016 seconds)

## < 4 Angle in 3D space Calculators

Angle between two planes
angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))) Go
Angle made by direction cosines of two lines in sine form
angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2)) Go
Angle between two lines given direction ratios of two lines
angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2))) Go
Angle between line and plane given coefficients of line and plane
angle_a = asin(((Direction ratio 1 of line1*Direction Ratio 1)+(Direction ratio 2 of line1*Direction Ratio 2)+(Direction ratio 3 of line1*Direction Ratio 3))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction Ratio 1)^2+(Direction Ratio 2)^2+(Direction Ratio 3)^2))) Go

### Angle between two planes Formula

angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2)))
∠C = acos(((l1*l2)+(m1*m2)+(n1*n2))/(sqrt((l1)^2+(m1)^2+(n1)^2)* sqrt((l2)^2+(m2)^2+(n2)^2)))

## What are direction cosines?

Direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction.

## How to Calculate Angle between two planes?

Angle between two planes calculator uses angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))) to calculate the Angle C, The Angle between two planes formula is defined as as the angle between the normal to them from any point. Angle C and is denoted by ∠C symbol.

How to calculate Angle between two planes using this online calculator? To use this online calculator for Angle between two planes, enter Direction cosine 1 with respect to x axis (l1), Direction cosine 2 with respect to x axis (l2), Direction cosine 1 with respect to y axis (m1), Direction cosine 2 with respect to y axis (m2), Direction cosine 1 with respect to z axis (n1) & Direction cosine 2 with respect to z axis (n2) and hit the calculate button. Here is how the Angle between two planes calculation can be explained with given input values -> 7.893159 = acos(((0.7*0.7)+(0.8*0.6)+(0.6*0.6))/(sqrt((0.7)^2+(0.8)^2+(0.6)^2)* sqrt((0.7)^2+(0.6)^2+(0.6)^2))).

### FAQ

What is Angle between two planes?
The Angle between two planes formula is defined as as the angle between the normal to them from any point and is represented as ∠C = acos(((l1*l2)+(m1*m2)+(n1*n2))/(sqrt((l1)^2+(m1)^2+(n1)^2)* sqrt((l2)^2+(m2)^2+(n2)^2))) or angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))). Direction cosine 1 with respect to x axis is the cosine of angle made by a line w.r.to x axis, Direction cosine 2 with respect to x axis is the cosine of angle made by a line w.r.to x axis, Direction cosine 1 with respect to y axis is the cosine of angle made by a line w.r.to y axis, Direction cosine 2 with respect to y axis is the cosine of angle made by a line w.r.to y axis, Direction cosine 1 with respect to z axis is the cosine of angle made by a line w.r.to z axis & Direction cosine 2 with respect to z axis is the cosine of angle made by a line w.r.to z axis.
How to calculate Angle between two planes?
The Angle between two planes formula is defined as as the angle between the normal to them from any point is calculated using angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))). To calculate Angle between two planes, you need Direction cosine 1 with respect to x axis (l1), Direction cosine 2 with respect to x axis (l2), Direction cosine 1 with respect to y axis (m1), Direction cosine 2 with respect to y axis (m2), Direction cosine 1 with respect to z axis (n1) & Direction cosine 2 with respect to z axis (n2). With our tool, you need to enter the respective value for Direction cosine 1 with respect to x axis, Direction cosine 2 with respect to x axis, Direction cosine 1 with respect to y axis, Direction cosine 2 with respect to y axis, Direction cosine 1 with respect to z axis & Direction cosine 2 with respect to z axis and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Angle C?
In this formula, Angle C uses Direction cosine 1 with respect to x axis, Direction cosine 2 with respect to x axis, Direction cosine 1 with respect to y axis, Direction cosine 2 with respect to y axis, Direction cosine 1 with respect to z axis & Direction cosine 2 with respect to z axis. We can use 4 other way(s) to calculate the same, which is/are as follows -
• angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2))
• angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2)))
• angle_a = asin(((Direction ratio 1 of line1*Direction Ratio 1)+(Direction ratio 2 of line1*Direction Ratio 2)+(Direction ratio 3 of line1*Direction Ratio 3))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction Ratio 1)^2+(Direction Ratio 2)^2+(Direction Ratio 3)^2)))
• angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2)))
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