# radius = distance? arc length = height?

as the title said, I have a little trouble to find which is radius and arc length

1)If a hill 2500 ft away subtend at 1.5 degree angle, how high is it?

my thinking: it ask for the arc length(height), and radius(distance) = 2500, angle = 1.5

2)At what distance does a tree 24 ft tall subtend an angle of 10'?

my thinking: it ask for the radius(distance), arc length(height) = 25, angle = 10'

so to the two questions above, my thinking is: distance is radius, height is arc length, and I calculated and it is correct, but while working on the question below, then I am confuse

3)An 18-in.mallard duck files overhead, subtending an angle of 3 degree 15'. how high is this duck flying?

my thinking:it ask for the height, radius = 18-in(1.5 ft), and the degree = 3 degree 15'(0.05669)

so i use the formula: s = rΘ = (1.5)(0.05669) = 0.085035ft , but it is wrong, the answer is 26ft which using the formula: r = s/Θ = 1.5/0.05669 = 26ft (rounded)

so i am thinking it is true that arc length = height, and radius = distance? or both would switch base on different problems?

thank

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Roughly speaking, in the formula $s=r\theta$, the symbol $s$ is the size (length) of the object you are observing, and $r$ is the distance from your eye to the object.
So in the mallard question, we have $s=18''$, and "how high it is flying" is $r$. Thus $r=\dfrac{s}{\theta}$. since you will want to give the answer in feet, you can do one of two things: (i) Set $s=18$, and compute $r$. The answer will be in inches, and you convert to feet by dividing by $12$; or (ii) Convert the $18''$ to feet ($1.5$) and compute.
For the hill, the distance you are from it ("$r$") is known, and you want $s$, the size of the hill. We use the formula $s=r\theta$. We know $r$, and $\theta$, so we multiply.
Well, they don't really switch. In $s=r\theta$, the $r$ always means "how far away, radius" and $s$ always means "how big, arc length." It is usually not hard to figure out what is what in a practical problem. In the hill problem, how far away is $2500$, and we want the size. In the tree problem, we don't know how far away, but we know the size. In the duck problem, again we don't know how far away, but we know the size. – André Nicolas Mar 5 '13 at 2:00