Question

Astronomy: At is closest approach, Mars is about 5.6X10^7 km from Earth and its apparent size is about 0.00012 radians. What is the approximate diameter of Mars?

Answer 1

6700 km  
For this problem, there are two basic ways of solving it and both will give approximately the same answer. The easiest way is to simply ignore the slight difference in length of the adjacent leg of the right triangle and hypotenuse and calculate the arc tangent. The equation is 
tan θ = opp/adj 
adj * tan θ = opp 
5.6x10^7 km *0.000120000000576 = opp 
6.720000032256x10^3 = opp 
6720 km = opp  
So the diameter of Mars is about 6700 km.  
The other way is to actually acknowledge that the hypotenuse and adjacent leg are different lengths and divide the apparent size by 2 for performing the trig, then double the result. So: 
tan θ = opp/adj 
adj * tan θ = opp 
5.6x10^7 km * 0.000060000000072 = opp
 5.6x10^7 km * 0.000060000000072 = opp 
3.360000004032x10^3 = opp
 3.360000004032x10^3 * 2 = diameter
 6.720000008064x10^3 = diameter 
 And you may notice the digits at the end of the results do differ. But they're so insignificant, that the final results in both cases is the same. So to 2 significant digits, the diameter of Mars is 6700 km.

Answer 2

1.) D = 2pi * d * A / 360 degrees; Where:
   pi = 3.14;
   d = distance or 56,000,000 km (5.6x10^7 to standard notation)
   A = angular size in degrees or .00688 degrees (.00012 radian to degrees)

  D = 2(3.14) * 56,000,000 * .00688 / 360
      = 6.28 * 385,280 / 360
      = 2,419,558.4 / 360
 D  = 6,720.996 or 6,721

Therefore, 6,721 km. is the approximate diameter of Mars.