How can the radius of a star be measured using parallax?

Question

The radius of Betelgeuse was measured using parallax. From the Wikipedia article:

the estimated parallax from the Hipparcos mission was 7.63±1.64 mas, yielding an estimated radius for Betelgeuse of 3.6 AU.

I understand how a distance can be determined using parallax, but I don't understand how the radius can.

What is the math behind deriving a radius from a parallax measurement?

Answer 1

Betelgeuse is large and close enough that it's angular diameter can be measured directly (via optical interferometry etc...)

When you have the angular diameter, knowing the distance lets you calculate the radius with simple trigonometry.

The fact that Betelgeuse has such a large angular diameter has actually made parallax measurements more difficult, because you are measuring the relative movement of a disc that's not insignificant in angular diameter compared to its angular parallax movement, instead of a single point. So the distance to Betelgeuse has been more uncertain than might normally be expected from a relatively nearby star.

Answer 2

The answer lies in the sentence immediately before the one you quoted an excerpt of.

In a study published in December 2000, the star's diameter was measured with the Infrared Spatial Interferometer (ISI) at mid-infrared wavelengths producing a limb-darkened estimate of 55.2±0.5 mas—a figure entirely consistent with Michelson's findings eighty years earlier.^[31][50] At the time of its publication, the estimated parallax from the Hipparcos mission was 7.63±1.64 mas, yielding an estimated radius for Betelgeuse of 3.6 AU.

(emphasis mine)

You combine the angular diameter with the distance obtained from the parallax to get the physical diameter.

$$\frac{r}{\mathrm{1\ AU}} = \frac{D_\theta}{2\varpi}$$

Where $r$ is the physical radius, $D_\theta$ is the angular diameter, and $\varpi$ is the parallax. The factor of 2 comes from conversion of diameter to radius.