Cut a tiny square out of a loaded part and you can describe everything happening to it with three numbers: $\sigma_x$, the normal stress on the x face; $\sigma_y$, the normal stress on the y face; and $\tau_{xy}$, the shear stress acting along those faces (tau, the Greek t). All three in MPa, tension positive. Now rotate that square. The three numbers change — and somewhere in the rotation the shear disappears entirely, leaving two pure normal stresses. Those two are the principal stresses, and they are the largest and smallest normal stress the point will ever see on any plane.

Otto Mohr noticed that as you rotate, the pair traces a circle. So the arithmetic is just a centre and a radius. The centre is the average normal stress, $\tfrac{1}{2}(\sigma_x + \sigma_y)$. The radius is $R = \sqrt{\left(\dfrac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$ — R equals the root of half the difference squared plus tau squared. Note the DIFFERENCE inside the root, never the sum; that is the most-missed detail on this page. Then $\sigma_1 = \text{centre} + R$ and $\sigma_2 = \text{centre} - R$, and σ₂ is often negative, which simply means compression.

Here is the nugget that makes the circle worth drawing. The maximum in-plane shear stress $\tau_{max}$ is the radius itself — the same R, no extra work: $\tau_{max} = R = \dfrac{\sigma_1 - \sigma_2}{2}$. One quantity, two names. Ductile metals fail in shear, so τmax is very often the number that decides the part, and you get it free the moment you have the circle.