Angle of Deviation Formula: δ = i + e − A for a Prism

The angle of deviation formula for a prism is $\delta = i + e - A$ , where i is the angle of incidence, e the angle of emergence and A the angle of the prism. The deviation δ is the angle between the direction of the ray going in and the direction of the ray coming out. A ray that enters at 50° and leaves at 47° through a 60° prism has been turned through 37°.

Inside the prism the two refraction angles always add up to the prism angle, $r_1 + r_2 = A$ . At one special angle of incidence the deviation is as small as it can be, and that minimum deviation gives the refractive index of the glass. For a thin prism the whole story shrinks to $\delta = (n - 1)A$ . All three results are Class 12 ray optics staples and appear in JEE and NEET questions on prisms.

Quick facts

Prism formula
$\delta = i + e - A$
Angles inside
$r_1 + r_2 = A$
Minimum deviation
$n = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$
Thin prism
$\delta = (n - 1)A$ , A in degrees or radians
Unit
degree (°) or radian (rad)
Dimensional formula
[M⁰ L⁰ T⁰]
At minimum deviation
i = e and r₁ = r₂ = A/2; the ray passes symmetrically

Formula of angle of deviation and its symbols

Angles at a triangular prism
Symbol	Angle	Measured from
$\delta$	Angle of deviation	Incident ray direction to emergent ray direction
$i$	Angle of incidence at the first face	Normal to the first face
$e$	Angle of emergence at the second face	Normal to the second face
$A$	Angle of the prism (refracting angle)	Between the two refracting faces
$r_1, r_2$	Refraction angles inside the glass	Normals to each face

The ray bends toward the base of the prism at both faces. The deviation at the first face is $i - r_1$ and at the second face $e - r_2$ . Their sum is the total deviation.

Deriving the angle of deviation formula

Add the two deviations: $\delta = (i - r_1) + (e - r_2) = (i + e) - (r_1 + r_2)$ .
The two normals and the two faces form a four sided figure. Two of its corners, where the normals meet the faces, are right angles, so the angle between the normals is $180^\circ - A$ .
In the triangle formed by the ray inside the glass and the two normals, $r_1 + r_2 + (180^\circ - A) = 180^\circ$ , so $r_1 + r_2 = A$ .
Substitute into step 1: $\delta = i + e - A$ .

This is pure geometry: it does not use Snell's law at all. Snell's law enters when you need e from i, through $\sin i = n \sin r_1$ at the first face and $n \sin r_2 = \sin e$ at the second.

Minimum deviation and the refractive index

Plot δ against i and you get a U shaped curve. At its lowest point the ray passes through the prism symmetrically: $i = e$ and $r_1 = r_2 = \frac{A}{2}$ . Putting these into $\delta = i + e - A$ gives $i = \frac{A + \delta_m}{2}$ . Snell's law at the first face then gives the prism formula:

$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

A spectrometer reading of minimum deviation is how school labs measure the refractive index of a glass sample; the index of refraction page lists other routes. Because n is a little larger for violet than for red, violet deviates more and white light spreads into a spectrum.

Worked examples

Example 1: finding the deviation

A ray enters a 60° glass prism at i = 50° and leaves at e = 47.2°. Find the angle of deviation.

$\delta = 50^\circ + 47.2^\circ - 60^\circ = 37.2^\circ$

These numbers match glass of n = 1.50: sin r₁ = sin 50° / 1.5 gives r₁ = 30.7°, so r₂ = 29.3° and sin e = 1.5 sin 29.3° gives e = 47.2°.

Example 2: refractive index from minimum deviation

An equilateral prism (A = 60°) gives a minimum deviation of 30°. Find n.

$n = \frac{\sin 45^\circ}{\sin 30^\circ} = \frac{0.7071}{0.5} = 1.41$

Example 3: a thin prism in radians

A thin prism of angle 5° is made of glass with n = 1.5. Find the deviation in degrees and in radians.

$\delta = (n - 1)A = 0.5 \times 5^\circ = 2.5^\circ$

In radians: 2.5 × π/180 = 0.044 rad. The thin prism formula only works when A is small, roughly below 10°.

Enter angles in degrees in the calculator on this page. It solves δ = i + e − A for any one of the four angles.

Deviation in other situations

Angle of deviation outside a prism
Situation	Deviation
Refraction at a single surface	$\delta = i - r$
Reflection from a plane mirror	$\delta = 180^\circ - 2i$
Glass slab with parallel faces	Zero: the ray only shifts sideways
Two thin prisms in contact	$\delta = (n_1 - 1)A_1 + (n_2 - 1)A_2$

The slab result is a good check of the prism formula: parallel faces mean A = 0 and e = i, so δ = 0. Lenses can be pictured as stacks of tiny prisms, which is one way to see why the lens maker formula depends on n − 1. More optics is gathered on the waves and optics hub.

Exam slips with prism questions

Measuring i or e from the face instead of the normal.
Writing δ = i + e + A. The prism angle is subtracted.
Using the thin prism result for a 60° prism. It is only for small angles.
Forgetting that minimum deviation needs i = e. You cannot use the n formula with an arbitrary δ.
Halving the wrong angle. In the n formula both A and A + δm are halved.

Frequently asked questions

What is the formula of angle of deviation?

For a prism, δ = i + e − A, where i is the angle of incidence, e the angle of emergence and A the prism angle. For a thin prism, δ = (n − 1)A.

What is the angle of minimum deviation?

It is the smallest deviation a prism can produce, reached when the ray passes through symmetrically with i = e. It is used to find the refractive index with n = sin((A + δm)/2) / sin(A/2).

Is the deviation different for red and violet light?

It is. Violet light sees a slightly higher refractive index than red, so it is deviated more. This difference is dispersion.

What is the SI unit of angle of deviation?

The radian, though degrees are used in almost every prism problem. As a ratio of arc length to radius, an angle has no dimensions.

Why is r₁ + r₂ = A in a prism?

The normals to the two faces meet at an angle of 180° − A. The angles in the triangle formed by the ray inside the glass and the two normals add to 180°, which leaves r₁ + r₂ = A.