Angle of Incidence Calculator
Calculate angle of incidence for light rays using surface normal and incident ray direction vectors.
What Is the Angle of Incidence?
The angle of incidence is the angle between an incoming light ray and the surface normal at the point where the ray strikes a boundary between media.
Angle of Incidence Formula
$$\theta_i = \arccos\left(-\hat{i} \cdot \hat{n}\right)$$Here $\hat{i}$ is the unit vector along the incident ray (direction of propagation) and $\hat{n}$ is the outward surface normal. The negative sign accounts for the ray traveling toward the surface.
Using Angles or Coordinates
Enter ray and normal angles measured counterclockwise from the +x axis, or provide 2D direction vectors directly. Related tools: Angle of Refraction Calculator and Snell's Law Calculator.
Frequently Asked Questions
What is the surface normal?
The surface normal is a vector perpendicular to the surface at the point of contact, pointing outward from the medium.
Why is there a negative sign in the formula?
The incident ray direction points along propagation. Negating it aligns the vector with the approach direction relative to the outward normal.
What range should the angle be?
The angle of incidence is between 0° (normal incidence) and 90° (grazing incidence).
How does this relate to refraction?
Snell's law uses the angle of incidence together with refractive indices to determine the refracted angle in the second medium.
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