The moon's distance from the earth is 360000 km and its diameter subtends an angle of 31 minute at the of the observer on the earth . Find the radius of the moon.

Make a sketch, (it won't be to scale!)

You will have an isosceles triangle with an angle at the earth of 31 minutes of angle.
Draw in a perpendicular which would be
360,000 km
If the radius of the moon is r, I see it as
tan (0°15'30") = r/360000
r = 360000(tan(0°15'30") ) = 1623 km

= appr 1600 km

Google states the radius as 1737 km

PS,
on my calculator I have a key labeled
D°M'S" which I used for my angle above,
you could also have
31 minutes = 31/60 °
then half would be 31/120°

To find the radius of the moon, we can use the concept of similar triangles. First, let's convert the angle from minutes to degrees:

1 minute = 1/60 degrees

So, the angle subtended by the diameter of the moon can be expressed as:

31 minute * (1/60 degrees/1 minute) = 31/60 degrees

Now, let's consider a right triangle formed by the line connecting the observer on Earth, the center of the Earth, and the center of the Moon. The angle subtended at the observer by the diameter of the Moon is the same as the angle subtended at the center of the Earth by the radius of the Moon.

Let's denote the radius of the Moon as R and the distance between the observer on Earth and the Moon as D.

R / D = tan(angle)

Since we know the angle (31/60 degrees) and the distance (360,000 km), we can determine the radius of the Moon.

R / 360,000 km = tan(31/60 degrees)

Now, we need to rearrange the equation to solve for R:

R = 360,000 km * tan(31/60 degrees)

Calculating this expression will give us the value of the radius of the Moon.