## Interplanar Angle for Simple Cubic System Solution

STEP 0: Pre-Calculation Summary
Formula Used
Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2))))
θ = acos(((h1*h2)+(k1*k2)+(l1*l2))/(sqrt((h1^2)+(k1^2)+(l1^2))*sqrt((h2^2)+(k2^2)+(l2^2))))
This formula uses 3 Functions, 7 Variables
Functions Used
cos - Trigonometric cosine function, cos(Angle)
acos - Inverse trigonometric cosine function, acos(Number)
sqrt - Square root function, sqrt(Number)
Variables Used
Interplanar Angle - (Measured in Radian) - The Interplanar Angle is the angle, f between two planes, (h1, k1, l1) and (h2, k2, l2).
Miller Index along plane 1 - The Miller Index along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 1.
Miller Index h along plane 2 - The Miller Index h along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 2.
Miller Index k along Plane 1 - The Miller Index k along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 1.
Miller Index k along Plane 2 - The Miller Index k along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 2.
Miller Index l along plane 1 - The Miller Index l along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 1.
Miller Index l along plane 2 - The Miller Index l along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 2.
STEP 1: Convert Input(s) to Base Unit
Miller Index along plane 1: 5 --> No Conversion Required
Miller Index h along plane 2: 8 --> No Conversion Required
Miller Index k along Plane 1: 3 --> No Conversion Required
Miller Index k along Plane 2: 6 --> No Conversion Required
Miller Index l along plane 1: 16 --> No Conversion Required
Miller Index l along plane 2: 25 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
θ = acos(((h1*h2)+(k1*k2)+(l1*l2))/(sqrt((h1^2)+(k1^2)+(l1^2))*sqrt((h2^2)+(k2^2)+(l2^2)))) --> acos(((5*8)+(3*6)+(16*25))/(sqrt((5^2)+(3^2)+(16^2))*sqrt((8^2)+(6^2)+(25^2))))
Evaluating ... ...
θ = 0.0480969557269001
STEP 3: Convert Result to Output's Unit
0.0480969557269001 Radian -->2.75575257057947 Degree (Check conversion here)
2.75575257057947 2.755753 Degree <-- Interplanar Angle
(Calculation completed in 00.004 seconds)
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## Credits

Created by Prerana Bakli
National Institute of Technology (NIT), Meghalaya
Prerana Bakli has created this Calculator and 800+ more calculators!
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K J Somaiya College of science (K J Somaiya), Mumbai
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## < 10+ Inter-planar distance and inter-planar angle Calculators

Interplanar Distance in Triclinic Crystal Lattice
Interplanar Spacing = sqrt(1/((((Lattice Constant b^2)*(Lattice Constant c^2)*((sin(Lattice parameter alpha))^2)*(Miller Index along x-axis^2))+((Lattice Constant a^2)*(Lattice Constant c^2)*((sin(Lattice Parameter Beta))^2)*(Miller Index along y-axis^2))+((Lattice Constant a^2)*(Lattice Constant b^2)*((sin(Lattice Parameter gamma))^2)*(Miller Index along z-axis^2))+(2*Lattice Constant a*Lattice Constant b*(Lattice Constant c^2)*((cos(Lattice parameter alpha)*cos(Lattice Parameter Beta))-cos(Lattice Parameter gamma))*Miller Index along x-axis*Miller Index along y-axis)+(2*Lattice Constant b*Lattice Constant c*(Lattice Constant a^2)*((cos(Lattice Parameter gamma)*cos(Lattice Parameter Beta))-cos(Lattice parameter alpha))*Miller Index along z-axis*Miller Index along y-axis)+(2*Lattice Constant a*Lattice Constant c*(Lattice Constant b^2)*((cos(Lattice parameter alpha)*cos(Lattice Parameter gamma))-cos(Lattice Parameter Beta))*Miller Index along x-axis*Miller Index along z-axis))/(Volume of Unit Cell^2)))
Interplanar Angle for Hexagonal System
Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(0.5*((Miller Index along plane 1*Miller Index k along Plane 2)+(Miller Index h along plane 2*Miller Index k along Plane 1)))+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt(((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index along plane 1*Miller Index k along Plane 1)+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*(Miller Index l along plane 1^2)))*((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index h along plane 2*Miller Index k along Plane 2)+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*(Miller Index l along plane 2^2))))))
Interplanar Angle for Orthorhombic System
Interplanar Angle = acos((((Miller Index along plane 1*Miller Index h along plane 2)/(Lattice Constant a^2))+ ((Miller Index l along plane 1*Miller Index l along plane 2)/(Lattice Constant c^2))+ ((Miller Index k along Plane 1*Miller Index k along Plane 2)/(Lattice Constant b^2)))/ sqrt((((Miller Index along plane 1^2)/(Lattice Constant a^2))+((Miller Index k along Plane 1^2)/(Lattice Constant b^2))*((Miller Index l along plane 1^2)/(Lattice Constant c^2)))* (((Miller Index h along plane 2^2)/(Lattice Constant a^2))+((Miller Index k along Plane 1^2)/(Lattice Constant b^2))+((Miller Index l along plane 1^2)/(Lattice Constant c^2)))))
Interplanar Distance in Rhombohedral Crystal Lattice
Interplanar Spacing = sqrt(1/(((((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))*(sin(Lattice parameter alpha)^2))+(((Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis*Miller Index along z-axis)+(Miller Index along x-axis*Miller Index along z-axis))*2*(cos(Lattice parameter alpha)^2))-cos(Lattice parameter alpha))/(Lattice Constant a^2*(1-(3*(cos(Lattice parameter alpha)^2))+(2*(cos(Lattice parameter alpha)^3))))))
Interplanar Distance in Monoclinic Crystal Lattice
Interplanar Spacing = sqrt(1/((((Miller Index along x-axis^2)/(Lattice Constant a^2))+(((Miller Index along y-axis^2)*(sin(Lattice Parameter Beta)^2))/(Lattice Constant b^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))-(2*Miller Index along x-axis*Miller Index along z-axis*cos(Lattice Parameter Beta)/(Lattice Constant a*Lattice Constant c)))/((sin(Lattice Parameter Beta))^2)))
Interplanar Angle for Simple Cubic System
Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2))))
Interplanar Distance in Hexagonal Crystal Lattice
Interplanar Spacing = sqrt(1/((((4/3)*((Miller Index along x-axis^2)+(Miller Index along x-axis*Miller Index along y-axis)+(Miller Index along y-axis^2)))/(Lattice Constant a^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
Interplanar Distance in Orthorhombic Crystal Lattice
Interplanar Spacing = sqrt(1/(((Miller Index along x-axis^2)/(Lattice Constant a^2))+((Miller Index along y-axis^2)/(Lattice Constant b^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
Interplanar Distance in Tetragonal Crystal Lattice
Interplanar Spacing = sqrt(1/((((Miller Index along x-axis^2)+(Miller Index along y-axis^2))/(Lattice Constant a^2))+((Miller Index along z-axis^2)/(Lattice Constant c^2))))
Interplanar Distance in Cubic Crystal Lattice
Interplanar Spacing = Edge Length/sqrt((Miller Index along x-axis^2)+(Miller Index along y-axis^2)+(Miller Index along z-axis^2))

## Interplanar Angle for Simple Cubic System Formula

Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2))))
θ = acos(((h1*h2)+(k1*k2)+(l1*l2))/(sqrt((h1^2)+(k1^2)+(l1^2))*sqrt((h2^2)+(k2^2)+(l2^2))))

## What are Bravais Lattices?

Bravais Lattice refers to the 14 different 3-dimensional configurations into which atoms can be arranged in crystals. The smallest group of symmetrically aligned atoms which can be repeated in an array to make up the entire crystal is called a unit cell.
There are several ways to describe a lattice. The most fundamental description is known as the Bravais lattice. In words, a Bravais lattice is an array of discrete points with an arrangement and orientation that look exactly the same from any of the discrete points, that is the lattice points are indistinguishable from one another.
Out of 14 types of Bravais lattices some 7 types of Bravais lattices in three-dimensional space are listed in this subsection. Note that the letters a, b, and c have been used to denote the dimensions of the unit cells whereas the letters 𝛂, 𝞫, and 𝝲 denote the corresponding angles in the unit cells.

## How to Calculate Interplanar Angle for Simple Cubic System?

Interplanar Angle for Simple Cubic System calculator uses Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2)))) to calculate the Interplanar Angle, The Interplanar angle for Simple Cubic system is the angle between two planes, (h1, k1, l1) and (h2, k2, l2) in a Simple Cubic system. Interplanar Angle is denoted by θ symbol.

How to calculate Interplanar Angle for Simple Cubic System using this online calculator? To use this online calculator for Interplanar Angle for Simple Cubic System, enter Miller Index along plane 1 (h1), Miller Index h along plane 2 (h2), Miller Index k along Plane 1 (k1), Miller Index k along Plane 2 (k2), Miller Index l along plane 1 (l1) & Miller Index l along plane 2 (l2) and hit the calculate button. Here is how the Interplanar Angle for Simple Cubic System calculation can be explained with given input values -> 157.893 = acos(((5*8)+(3*6)+(16*25))/(sqrt((5^2)+(3^2)+(16^2))*sqrt((8^2)+(6^2)+(25^2)))).

### FAQ

What is Interplanar Angle for Simple Cubic System?
The Interplanar angle for Simple Cubic system is the angle between two planes, (h1, k1, l1) and (h2, k2, l2) in a Simple Cubic system and is represented as θ = acos(((h1*h2)+(k1*k2)+(l1*l2))/(sqrt((h1^2)+(k1^2)+(l1^2))*sqrt((h2^2)+(k2^2)+(l2^2)))) or Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2)))). The Miller Index along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 1, The Miller Index h along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the x-direction in plane 2, The Miller Index k along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 1, The Miller Index k along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the y-direction in plane 2, The Miller Index l along plane 1 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 1 & The Miller Index l along plane 2 form a notation system in crystallography for planes in crystal (Bravais) lattices along the z-direction in plane 2.
How to calculate Interplanar Angle for Simple Cubic System?
The Interplanar angle for Simple Cubic system is the angle between two planes, (h1, k1, l1) and (h2, k2, l2) in a Simple Cubic system is calculated using Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index l along plane 1^2))*sqrt((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index l along plane 2^2)))). To calculate Interplanar Angle for Simple Cubic System, you need Miller Index along plane 1 (h1), Miller Index h along plane 2 (h2), Miller Index k along Plane 1 (k1), Miller Index k along Plane 2 (k2), Miller Index l along plane 1 (l1) & Miller Index l along plane 2 (l2). With our tool, you need to enter the respective value for Miller Index along plane 1, Miller Index h along plane 2, Miller Index k along Plane 1, Miller Index k along Plane 2, Miller Index l along plane 1 & Miller Index l along plane 2 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 Interplanar Angle?
In this formula, Interplanar Angle uses Miller Index along plane 1, Miller Index h along plane 2, Miller Index k along Plane 1, Miller Index k along Plane 2, Miller Index l along plane 1 & Miller Index l along plane 2. We can use 2 other way(s) to calculate the same, which is/are as follows -
• Interplanar Angle = acos((((Miller Index along plane 1*Miller Index h along plane 2)/(Lattice Constant a^2))+ ((Miller Index l along plane 1*Miller Index l along plane 2)/(Lattice Constant c^2))+ ((Miller Index k along Plane 1*Miller Index k along Plane 2)/(Lattice Constant b^2)))/ sqrt((((Miller Index along plane 1^2)/(Lattice Constant a^2))+((Miller Index k along Plane 1^2)/(Lattice Constant b^2))*((Miller Index l along plane 1^2)/(Lattice Constant c^2)))* (((Miller Index h along plane 2^2)/(Lattice Constant a^2))+((Miller Index k along Plane 1^2)/(Lattice Constant b^2))+((Miller Index l along plane 1^2)/(Lattice Constant c^2)))))
• Interplanar Angle = acos(((Miller Index along plane 1*Miller Index h along plane 2)+(Miller Index k along Plane 1*Miller Index k along Plane 2)+(0.5*((Miller Index along plane 1*Miller Index k along Plane 2)+(Miller Index h along plane 2*Miller Index k along Plane 1)))+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*Miller Index l along plane 1*Miller Index l along plane 2))/(sqrt(((Miller Index along plane 1^2)+(Miller Index k along Plane 1^2)+(Miller Index along plane 1*Miller Index k along Plane 1)+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*(Miller Index l along plane 1^2)))*((Miller Index h along plane 2^2)+(Miller Index k along Plane 2^2)+(Miller Index h along plane 2*Miller Index k along Plane 2)+((3/4)*((Lattice Constant a^2)/(Lattice Constant c^2))*(Miller Index l along plane 2^2))))))
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