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Prism Deviation Calculator

Find the minimum deviation of a prism.

Result

37.18°

How it is calculated

The tool uses the minimum-deviation formula δ = 2 · asin( n · sin(A / 2) ) − A. A is the prism's apex angle in degrees, n is the refractive index of the glass relative to the surrounding air (taken as 1), and δ is the smallest angle by which the ray is turned, shown in degrees with two decimals.

Inputs outside physical limits give no result: the apex angle must be above 0° and below 180°, and the index must be at least 1. If n · sin(A / 2) is greater than 1 the ray is totally internally reflected inside the prism and cannot emerge symmetrically, so the tool again returns nothing; with A = 60°, for example, an index of 2.1 fails. The formula holds for light of a single colour, because n depends on wavelength.

Worked example

A prism has an apex angle of 60° and an index of 1.5. Half the apex angle is 30°, and sin 30° = 0.5. Then n · sin(A / 2) = 1.5 × 0.5 = 0.75. The arcsine of 0.75 is 48.59°, doubled that is 97.18°. Subtract the apex angle: 97.18° − 60° = 37.18°. The ray is turned by at least 37.18° at the minimum-deviation setting.

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Common mistakes

  • Entering the angle of incidence or the base angle in the apex field; A is the angle between the two faces the light passes through.
  • Typing the apex angle in radians, so 1.047 is read as about one degree instead of 60.
  • Using the index for the wrong colour: blue light has a higher n than red, so dispersive glass gives a different deviation for each wavelength.
  • Applying the result to any incident angle; the formula is only the minimum deviation, and other angles of incidence deviate the ray more.
  • Entering an index below 1 or a prism in water or oil without adjusting n; the tool assumes the prism sits in air, so use the relative index n(glass) / n(medium) when it does not.

About this calculator

When light passes through a triangular prism, it bends at both surfaces; at one specific angle of incidence the total deviation angle δ reaches a minimum, and at that symmetric point it relates cleanly to the prism's apex angle A and refractive index n through δ = 2·asin(n·sin(A/2)) − A — a rearrangement of Snell's law applied twice, once at each face.

This minimum-deviation condition is the standard method optics labs use to measure a material's refractive index precisely: by rotating a prism to find the angle of minimum deviation experimentally and then solving the formula for n, spectrometer measurements can determine refractive index to several decimal places without needing to know the light's exact path inside the glass.

Optics and photonics students verifying Snell's law in the lab, spectrometer operators calibrating prism instruments, and physics educators demonstrating dispersion all use this relationship, since it's one of the few refraction formulas that can be checked directly against a real, measurable prism.

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  • Editorial team

    The result is sensitive to the index, which is why labs measure angles carefully. With a 60° prism, raising n from 1.50 to 1.52 moves the minimum deviation from 37.18° to 38.93°, a change of about 1.75°. So a goniometer error of even a small fraction of a degree matters when you work backwards to n. The tool runs forward only. Do you want to check a lab measurement? Tell us your apex angle and index.

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