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Snell's Law Calculator

Find how much light bends between two materials: solve n₁ sin θ₁ = n₂ sin θ₂ for either angle or either index, with the critical angle.

Level: Grades 11–12

  • Common Core standard HSF.TF.B.7(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Printable worksheet

Snell's Law Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Solve for

Measured from the normal (the line perpendicular to the surface), not from the surface itself.

Result

Refraction angle (θ₂)

22.0369 °

The refracted ray leaves at 22.0369° from the normal.

Deviation (θ₁ − θ₂)
7.9631 °
Critical angle
None: total internal reflection can't occur going into an equal or higher index
Bending
Toward the normal

See the step-by-step solution ↓

Step-by-step solution

  1. Use an approximate refractive index for air

    n₁=1.000293

  2. Use an approximate refractive index for water

    n₂=1.333

  3. Rearrange Snell's law to find the sine of the bent (refraction) angle: sin θ₂ = n₁ × sin θ₁ ÷ n₂

    1.000293×sin 30°1.333=1.000293×0.51.333=0.37520368

  4. Turn the sine back into an angle with the inverse sine (arcsin)

    θ₂ = arcsin 0.37520368=22.036902 °

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how much a ray of light bends when it passes from one clear material into another. Light changes direction when it crosses from, say, air into water. That's why a straw looks bent in a glass of water, and why lenses can focus. Snell's law says exactly how much it bends: n₁ × sin θ₁ = n₂ × sin θ₂. Here n is each material's refractive index, a number for how much it slows light. θ (theta) is the angle of the ray. It's measured from the normal, an imaginary line at right angles to the surface.

This calculator solves the law for the bent (refraction) angle, the incoming (incidence) angle, or either refractive index. Sometimes light heads into a material with a lower index, like from water into air. Then it also gives the critical angle. It tells you if the ray bounces back entirely (total internal reflection) instead of passing through.

Where it's used

Why this math matters outside the classroom:

  • Fiber optics. Fiber-optic cables keep light inside by total internal reflection, which needs angles beyond the core's critical angle.
  • Shallow-looking pools. Swimmers see a pool look shallower than it is because light bends as it leaves the water.
  • Critical angle. Light going from glass with n = 1.5 into air has a critical angle of arcsin(1 ÷ 1.5) ≈ 41.8°.
  • Lens design. Eyeglass and camera lens designers trace how rays bend at each curved surface to bring them to a focus.
  • Air into water. A ray entering water (n = 1.33) from air at 30° bends to about arcsin(0.5 ÷ 1.33) ≈ 22.1°.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSF.TF.B.7. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose what to solve for: the refraction angle θ₂, the incidence angle θ₁, or the index n₁ or n₂.
  2. Pick the first material (where the light starts) and the second material (where it goes). Or enter your own refractive indices.
  3. Enter the known angle or angles in degrees, measured from the normal.
  4. Read the result. You'll also see the critical angle, if there is one, and whether the ray bends toward or away from the normal.

What the inputs mean

First medium (light comes from here)
Preset indices are approximate values for yellow light near room temperature. The exception is vacuum, which is exactly 1. Air is about 1.000293, water about 1.333, and ordinary glass about 1.5. Real glasses range from about 1.47 to over 1.9.
Angle of incidence (θ₁)
Angles are measured from the normal, not from the surface. A ray hitting the surface head-on has an angle of 0°. A ray skimming along the surface has 90°.
Angle of refraction (θ₂)
Also measured from the normal, on the far side of the surface.

Formula and method

n₁ × sin θ₁ = n₂ × sin θ₂
θ₂ = arcsin(n₁ × sin θ₁ ÷ n₂)
θc = arcsin(n₂ ÷ n₁)   (only when n₁ > n₂)
speed in a medium = c ÷ n

n₁ and θ₁ are the index and angle in the first material, and n₂ and θ₂ are the same for the second. sin is the sine of the angle, and arcsin turns a sine back into an angle. θc is the critical angle, and c is the speed of light in a vacuum.

The refractive index is how many times slower light travels in a material than in a vacuum. When light enters a slower (higher-index) material, it bends toward the normal. When it enters a faster (lower-index) material, it bends away from it.

Going into a lower index, the formula can ask for sin θ₂ greater than 1, and no angle has a sine that big. Then no light gets through, and it all reflects back. That's total internal reflection. The critical angle θc is the incoming angle at which the bent ray just skims the surface, at 90°.

Worked example

A ray of light in air hits a calm water surface at 30° from the normal. At what angle does it travel through the water?

  1. sin θ₂ = n₁ × sin θ₁ ÷ n₂ = 1.000293 × sin 30° ÷ 1.333 = 1.000293 × 0.5 ÷ 1.333 = 0.37520368.
  2. θ₂ = arcsin 0.37520368 = 22.0369°.

Result: The ray bends toward the normal. It travels through the water at 22.0369° from the normal, about 8° steeper than it arrived.

Understanding the result

If θ₂ is smaller than θ₁, the ray bent toward the normal, into a higher-index (slower) material. A larger θ₂ means it bent away from the normal.

If the result is total internal reflection, the surface acts as a perfect mirror for that ray. Optical fibers, prism binoculars and the sparkle of cut gemstones all rely on this.

When you solve for an index, a result below 1 is unusual for visible light. It often points to a mismatched angle or the wrong material.

Assumptions and limitations

  • The surface is flat and smooth, and both materials are uniform and transparent.
  • The light is a single color. Preset indices are for yellow light around 589 nm (nanometers, the light's wavelength).
  • Angles are measured from the normal, in degrees.
  • The refractive index changes slightly with color, an effect called dispersion. That's what splits white light in a prism. Use the index for your wavelength when precision matters.
  • The calculator only gives directions. It doesn't say how much light is reflected and how much passes through at a surface. That needs the Fresnel equations.
  • Birefringent crystals, such as calcite, have two refractive indices and need a more detailed approach.

Frequently asked questions

What is the critical angle?

When light goes from a higher index into a lower one, the critical angle is the incoming angle at which the bent ray runs along the surface. At any larger angle, all the light reflects. For water into air it's about 48.6°. For typical glass into air it's about 41.8°.

Why does a straw look bent in water?

Light from the underwater part of the straw bends away from the normal as it leaves the water. Your eyes trace the rays back in straight lines. So the underwater part looks higher and shifted, and the straw seems to bend at the surface.

Can total internal reflection happen going from air into water?

No. It only happens when light travels into a lower-index material. From air into water, the light bends toward the normal, and some always gets through, at any angle.

How do I measure a refractive index with this?

Shine a narrow beam from air into the material. Measure the incoming and bent angles from the normal, then solve for n₂. For example, 45° in and 28.13° inside gives about 1.5, typical of glass.

How this calculator is checked

Method: Snell's law of refraction n₁ sin θ₁ = n₂ sin θ₂; critical angle θc = arcsin(n₂/n₁). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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