How does Snell's Law relate angle of incidence to angle of refraction?

Snell's Law states that n1 times sin(theta1) equals n2 times sin(theta2), where n1 is the refractive index of the first medium, theta1 is the angle of incidence, n2 is the refractive index of the second medium, and theta2 is the angle of refraction. All angles are measured from the normal to the surface. If the second medium is denser (higher n), the refracted ray bends toward the normal and the angle of refraction is smaller than the angle of incidence.

What is the law of reflection for the angle of incidence?

The law of reflection states that the angle of incidence equals the angle of reflection. Both angles are measured from the normal to the reflecting surface. This law holds for all smooth (specular) surfaces, including mirrors and still water. The incident ray, the reflected ray, and the normal all lie in the same plane. This is distinct from refraction, which involves a ray passing into a second medium and bending according to Snell's Law.

Angle of Incidence Calculator

Calculate the angle of incidence, angle of refraction, or refractive index using Snell's Law. Choose what you want to solve for, enter the known values, and get instant results with a full breakdown.

The law of reflection also applies: the angle of incidence always equals the angle of reflection at a smooth surface.

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Verified June 2026 Standard optics formula. Snell's Law (Willebrord Snellius, 1621).

1. Solve For

Unknown to calculate

Medium 1 (incident medium)

Refractive index n1 (medium 1)

Medium 2 (refracted medium)

Refractive index n2 (medium 2)

2. Known Angles

Angle of incidence (theta1) degrees

deg

Angle of refraction (theta2) degrees

deg

Angle of incidence units

Please check your inputs. Angles must be between 0 and 90 degrees, and refractive indices must be greater than zero.

Tip: All angles are measured from the normal (the line perpendicular to the surface), not from the surface itself.

Results

Angle of Incidence (theta1)

From normal to surface

Angle of Refraction (theta2)
-
In second medium

Angle of Reflection

Equals angle of incidence

Critical Angle

For total internal reflection

Snell's Law Breakdown

Formula usedn1 sin(theta1) = n2 sin(theta2)

n1 (incident medium)-

theta1 (angle of incidence)-

n1 x sin(theta1)-

n2 (refracted medium)-

theta2 (angle of refraction)-

n2 x sin(theta2)-

Additional Results

Angle of reflection-

Ray bends toward normal?-

Critical angle (if n1 > n2)-

Total internal reflection?-

sin(theta1)-

sin(theta2)-

n ratio (n1 / n2)-

Result: Enter your values above to calculate.

What Is the Angle of Incidence?

The angle of incidence is the angle between an incoming ray (of light, sound, or another wave) and the normal to the surface at the point of contact. The normal is an imaginary line drawn perpendicular to the boundary between two media. The angle is always measured from the normal, not from the surface itself.

When a ray of light travels from one medium into another (for example, from air into water), it changes direction. This bending of light is called refraction. How much the ray bends depends on the refractive indices of both media and the angle at which the ray strikes the boundary.

Snell's Law

The relationship between the angle of incidence and the angle of refraction is described by Snell's Law:

n₁ sin(θ₁) = n₂ sin(θ₂)

Where:

n1 is the refractive index of the first (incident) medium
theta1 is the angle of incidence (measured from the normal)
n2 is the refractive index of the second (refracted) medium
theta2 is the angle of refraction (measured from the normal)

This calculator lets you solve for any one of the four values given the other three.

The Law of Reflection

Alongside refraction, some of the incident light is always reflected at the boundary. The law of reflection states that the angle of incidence equals the angle of reflection:

θ_incidence = θ_reflection

Both angles are measured from the normal, and all three rays (incident, reflected, refracted) lie in the same plane.

Total Internal Reflection and the Critical Angle

When light travels from a denser medium (higher n) to a less dense medium (lower n), there is a maximum angle of incidence beyond which no refraction occurs. All the light is reflected back into the first medium. This is called total internal reflection, and the minimum angle at which it occurs is the critical angle:

θ_c = arcsin(n₂ / n₁)

Total internal reflection is the principle behind optical fibres, periscopes, and diamond brilliance.

Common Refractive Indices

Medium	Refractive Index (n)
Vacuum	1.000 (exact)
Air (at standard conditions)	1.000293
Ice	1.309
Water (at 20 deg C)	1.333
Ethanol	1.361
Crown glass	1.500
Borosilicate glass	1.523
Diamond	2.417

Worked Example

A ray of light travels from air (n1 = 1.000) into water (n2 = 1.333) at an angle of incidence of 45 degrees. What is the angle of refraction?

Applying Snell's Law:

n1 sin(theta1) = n2 sin(theta2)
1.000 x sin(45 deg) = 1.333 x sin(theta2)
0.7071 = 1.333 x sin(theta2)
sin(theta2) = 0.7071 / 1.333 = 0.5304
theta2 = arcsin(0.5304) = 32.03 deg

The ray bends toward the normal because water is denser than air (n2 > n1). The angle of reflection at the air-water surface is also 45 degrees (law of reflection).

Related Calculators

Sources: Hecht, E. (2017). Optics (5th ed.). Pearson. Snell's Law derivation and refractive index values: NIST Physical Reference Data (physics.nist.gov). Refractive index database values: refractiveindex.info.

This calculator applies the standard Snell's Law formula for monochromatic light at a planar interface. Refractive indices vary with wavelength (dispersion); the values shown are approximate values for visible light at standard conditions. For precise optical engineering work, consult a full spectral data source.