Angle of Incidence - Snell's Law and Reflection

An angle of incidence calculator turns a measured ray angle into the four angles that define how light behaves when it meets a boundary. The angle of incidence is the angle between an incoming ray and the surface normal (the imaginary line perpendicular to the surface at the point of contact). That single number drives the angle of reflection, the angle of refraction through Snell's law, and the critical angle that marks the start of total internal reflection.

• • Physics homework and lab checks: Plug in a ray and two media to confirm reflection and refraction angles faster than working the algebra.
• • Optics and lens design: Compare angles of refraction across glass, water, and air when prototyping a prism or lens.
• • Solar and lighting siting: Estimate the angle at which sunlight strikes a panel or skylight.
• • Fiber optic and waveguide work: Compute the critical angle for a glass-to-air interface to confirm light stays trapped inside the core.

The angle is always measured from the normal, never from the surface itself. The same Snell's law relationship predicts whether the ray can escape the first medium at all, which is why total internal reflection matters.

Once the incidence and reflection angles are in hand, the same normal-based geometry drives image distances, so Mirror Equation Calculator applies these angles to find where a reflected image forms in a concave, convex, or plane mirror.

Behind the result panel sit two pieces of geometry and one well-known law of refraction. The calculator applies the law of reflection, then Snell's law, then a check against the critical angle.

• θ_surface: Angle between the ray and the surface, the number the user enters.
• θ_i: Angle of incidence measured from the normal. Equals 90° - θ_surface.
• θ_r: Angle of reflection. Equals θ_i by the law of reflection.
• n₁, n₂: Refractive indices of the two media.
• θ₂: Angle of refraction in medium 2, from Snell's law.
• θ_c: Critical angle, the smallest incidence that triggers total internal reflection.

The law of reflection does not depend on the second medium, so the reflected angle always equals the angle of incidence.

When the second medium is less dense than the first, Snell's law cannot produce a real angle past the critical angle.

According to Wikipedia (Snell's law), the law of refraction states n1 sin θ1 = n2 sin θ2, where n1 and n2 are the refractive indices and θ1 and θ2 are the angles measured from the normal.

Snell's law applied to a curved surface becomes the lens equation, so Thin Lens Equation Calculator builds on these refraction angles to find object distance, image distance, and focal length for a thin lens.

Four ideas do almost all the work in this calculator. Get them straight and the result panel makes sense for any pair of media.

These four ideas also drive related optics tools. Snell's law becomes the thin-lens equation when the two refracting surfaces sit close together.

Curved refracting surfaces need a curvature correction that pure Snell's law cannot supply, so Lensmaker's Equation Calculator uses the same refractive indices together with surface radii to give the focal length of a real lens.

Three inputs and four outputs. The default example loads with a 30° surface angle between air and water, the most common classroom setup.

1. 1 Enter the angle from the surface: Type the angle between the ray and the surface in degrees, between 0 and 90. The default of 30° is the textbook air-to-water setup.
2. 2 Set the refractive index of medium 1: Enter the refractive index of the medium the ray is leaving. Air is 1.000273, water is 1.333.
3. 3 Set the refractive index of medium 2: Enter the refractive index of the medium the ray is entering. Defaults to water at 1.333.
4. 4 Read the angle of incidence: The first row shows the incidence angle measured from the normal and updates as you type.
5. 5 Check refraction and total internal reflection: The next rows report the law of reflection angle, the Snell's law refraction angle, and whether total internal reflection applies.
6. 6 Reset for a fresh example: Press Reset to restore the default 30° air-to-water setup and try a new pair of media.

A physics student measures a laser striking a glass block at a 25° angle from the surface. Enter 25 for the surface angle, leave n1 at 1.000273, and change n2 to 1.52 for window glass. The angle of incidence row reads 65°, the reflection row reads 65°, and the refraction row reads about 37.18°.

When a problem statement gives the angle in gradians or radians instead of degrees, swap the readout first with Angle Converter so the calculator's trig functions receive degrees and the result lands where you expect it.

The calculator collapses three pieces of optics algebra into a single result panel. These are the practical reasons to reach for it.

• • Fast homework and lab checks: Returns reflection, refraction, and critical angles without redoing the Snell's law algebra.
• • Catch sign and unit mistakes early: Removes the most common error of mixing surface and normal angles.
• • See total internal reflection at a glance: The Yes/No flag turns the critical angle from a separate computation into a single readout.
• • Compare media side by side: Swap n2 between glass, water, and air to see how the refraction angle shifts.
• • Plan solar and lighting geometry: Use the surface angle as the tilt of a panel or window to find the incidence angle from the normal.

The calculator is best for quick checks and for teaching the geometry of optics. For solar and daylighting work, the resulting incidence angle plugs directly into solar declination formulas and tilted-surface irradiance models.

Snell's law is exact for ideal media, but real materials and measurement choices can shift the answer by a degree or more.

• • Snell's law assumes isotropic media and a flat interface, so anisotropic crystals like calcite need a separate treatment.
• • The result panel reports geometric angles but does not split the incoming light into reflected and transmitted intensities.

For most textbook and lab work these caveats are small. When the ray grazes the surface (incidence near 90°), Snell's law resolves to a real number but the ray is no longer entering the second medium.

According to Britannica, refraction is the change in direction of a wave passing from one medium to another, and Snell's law relates the angles and refractive indices of the two media.

According to The Physics Classroom, the law of reflection states that the angle of incidence equals the angle of reflection when both are measured from the normal.

Refraction also affects wave fronts rather than just rays, so Harmonic Wave Equation Calculator uses the same refractive index idea to find the speed, wavelength, and frequency of a wave in a new medium.