In a prism, the angle of deviation (δ) decreases with increase in the angle of incidence (i) up to a particular angle. This angle of incidence where the angle of deviation in a prism is minimum is called the minimum deviation position of the prism and that very deviation angle is known as the minimum angle of deviation (denoted by δ_min, D_λ, or D_m).

The angle of minimum deviation is related with the refractive index as:

${\displaystyle n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}}$

This is useful to calculate the refractive index of a material. Rainbow and halo occur at minimum deviation. Also, a thin prism is always set at minimum deviation.

In minimum deviation, the refracted ray in the prism is parallel to its base. In other words, the light ray is symmetrical about the axis of symmetry of the prism. Also, the angles of refractions are equal i.e. r₁ = r₂. The angle of incidence and angle of emergence equal each other (i = e). This is clearly visible in the graph below.

The formula for minimum deviation can be derived by exploiting the geometry in the prism. The approach involves replacing the variables in the Snell's law in terms of the Deviation and Prism Angles by making the use of the above properties.

From the angle sum of ${\textstyle \triangle OPQ}$,

${\displaystyle A+\angle OPQ+\angle OQP=180^{\circ }}$ ${\displaystyle \implies A=180^{\circ }-(90-r)-(90-r)}$ ${\displaystyle \implies r={\frac {A}{2}}}$