We can use Stefan's Law to relate the luminosity L, radius R, and temperature T of a star to each other. If we know the value of two of these quantities, we can calculate the value of the third.

Stefan's Law, as written above, makes it easy to determine the value of L if you know the values of R and T. But what if you know L and R, but not T, or if you know L and T, but not R? The easiest way to solve these type of questions is to rewrite Stefan's Law so the unknown variable is the only thing on the left hand side of the equation.

Consider the case where we know the values of L and T, and want to determine R. We first divide the equation by the fourth power of T (T × T × T × T) on both sides, so that all factors of L and T will be on one side and all factors of R will be alone on the other side of the equation.

To solve for R rather than the second power of R (R squared, or R × R), we take the square root of both sides of the equation. Now we have a equation that will tell us the value of R if we know L and T!

We next consider the case where we know the values of L and R, and want to determine T. We first divide the equation by the second power of R on both sides, so that all factors of L and R will be on one side and all factors of T will be alone on the other side of the equation.

To solve for T rather than the fourth power of T, we take the fourth root of both sides of the equation. Now we have a equation that will tell us the value of T if we know L and R!

Remember to express L, R, and T in solar units when using the equation. The values of L and R will always be given in terms of the solar luminosity and the solar radius, but you need to check whether you value for T is is units of kelvin (the Sun has a temperature of 5800 K), or in units of the solar temperature (where a star as hot as the Sun would have a temperature of 1 T_Sun).

Consider the case of a star with a temperature of 11600 K. What is its temperature in solar units? We simply rescale the temperature in terms of the solar value.

In the opposite case, what if we know that the star is three times hotter than the Sun. What is its temperature, in units of kelvin? We simply reverse the process.