The diffraction condition,

$$\vec{k}'-\vec{k} = \vec{G},$$

can be drawn as an isosceles triangle,

where the angle between $\vec{k}$ and $\vec{k}'$ is $2\theta$. A little trigonometry shows,

$$\frac{|\vec{G}_{hkl}|}{2} = |\vec{k}|\sin\theta.$$

Since $d_{hkl} = \frac{2\pi}{|\vec{G}_{hkl}|}$ and $\lambda = \frac{2\pi}{|\vec{k}|}$, this can also be written as,

$$\lambda = 2d_{hkl}\sin\theta.$$

Here $d_{hkl}$ is the distance between the netplanes. There is also a common derivation of this formula using the lattice planes.

X-rays travelling in the direction $\vec{k}$ strike lattice planes at an angle $\theta$ to the horizontal. The x-rays reflect off the lattice planes. For these reflections, the angle of incidence equals the angle of reflection. If we consider rays that reflect off two adjacent lattice planes as shown in the diagram above, the lower ray travels a longer distance $2d_{hkl}\sin\theta$ compared to the upper ray. The extra distance is drawn in blue in the diagram. These two rays will add constructively at the detector if this extra distance is equal to an integer $n$ number of wavelengths,

$$n\lambda = 2d_{hkl}\sin\theta .$$

This is the condition for Bragg diffraction and is almost the same formula as was presented above. The notation is a little confusing. In the Bragg formula, $d_{hkl}$ is the distance between lattice planes, and the indices $h$, $k$, and $l$ must have no common divisor. When the indices have no common divisor, they are called Miller indices. When the indices are allowed to have a common divisor, $d_{hkl}$ is the distance between netplanes, and $hkl$ are called Laue indices.