Question

When at closest approach to Earth, Planet X is observed to be 3.8 arcseconds in diameter. What is Planet X's diameter in meters?

Answer 1

There is no enough data. You need to know the distance to the planet.

Explanation:

You can derive an expression:

`r=l*tan(alpha/2)`, where r is the radius of the planet, l the distance to the planet, and alpha its angular width.

`alpha` is a very small angle, therefore in radians:

`tan(alpha)=alpha`

Passing arcseconds to radians_

`tan(alpha)~~((alpha/s)/(3600 s/(degree)))*((pi radians)/(180 degrees))`

`tan(3.8/2)~~(1.9/3600)*(pi/180)=9.2xx10^-6`

Now, imagine the distance is 50 million km (Mars or Venus can be at that distance):

`r=50xx10^9*9.2xx10^-6=460xx10^3 m`

The diameter will be 920 thousand meters. (Not Mars, neither Venus).