Incidence angle from Snell's law calculator

Enter the refractive index of the incident medium, the refractive index of the transmitted medium, and the desired refracted angle in degrees from the normal. The calculator returns the required incidence angle, intermediate sine values, and the relative index. It also identifies targets that no real incidence angle can produce, rather than returning an undefined or misleading result.

Rearranging Snell's law for the incidence angle

Snell's law is usually written as n1 times sine theta1 equals n2 times sine theta2. Here, n1 is the refractive index of the medium containing the incoming ray, theta1 is the incidence angle, n2 is the index of the medium the ray enters, and theta2 is the refraction angle. This calculator treats theta2 as the requested value and solves the equation in reverse. Dividing by n1 gives sine theta1 equals n2 divided by n1 times sine theta2. Taking the inverse sine then gives the required incidence angle: theta1 equals arcsine of that product. Every angle is measured in degrees from the surface normal. A ray aimed directly along the normal therefore has an angle of zero degrees, while a ray approaching parallel to the boundary tends toward ninety degrees. That convention matters because measuring from the surface would produce complementary values and an incorrect physical setup. The calculation assumes two homogeneous, isotropic media separated by one flat interface. It does not account for curved surfaces, graded-index material, wavelength-dependent dispersion, multilayer interference, or polarization effects. Within that scope, the result is deterministic and uses the same pure function in the browser and API.

Checking whether the requested refraction is physically reachable

An inverse calculation needs a domain check that can be easy to miss. The sine of a real angle cannot be greater than one, so the quantity n2 divided by n1 times sine theta2 must stay at or below one. If it exceeds one, no incidence angle between zero and ninety degrees can create the requested refracted ray for those two indices. This situation often appears when the transmitted medium has a larger refractive index than the incident medium and the requested refraction angle is too large. The calculator rejects that request with an invalid-input message and reports the maximum reachable refraction angle for context. At the exact boundary, the required incident ray is grazing at ninety degrees. Small floating-point noise near that boundary is safely clamped, while genuinely impossible values are never silently altered. Both refractive indices must be finite and strictly positive, because refractive index zero or a nonnumeric value does not define this form of Snell's law. The target refraction angle must be from zero through ninety degrees, inclusive. Use absolute refractive indices for ordinary optical work, such as approximately one for air, 1.333 for water, or about 1.5 for common glass.

Using the result in experiments, designs, and software

The returned angle_incidence value is the direction to set relative to the interface normal. In a classroom experiment, rotate the incoming laser or the sample until the incident ray has that angle, then compare the measured transmitted ray with the requested angle_refraction. In optical layout work, use the result as a first-order boundary calculation before moving to a full ray tracer that includes thickness, curvature, and multiple surfaces. Software can also use this capability to position an interactive ray, validate a proposed geometry, or generate answer keys from a list of materials and target angles. The response echoes the cleaned indices and target angle, provides both sine values for auditing, and returns n2 divided by n1 as relative_index. Those intermediate fields make it easier to verify unit conversions and catch a swapped medium order. The formula and unit fields are suitable for display, but the numeric fields should remain the source for downstream computation. A successful API request uses the published base price of $0.002 per item, while the browser calculator runs locally. Because the algorithm performs constant-time arithmetic without network access, repeated calls with identical inputs return identical values and are appropriate for automated tests.