Small angle approximation - Harvard University

SPU 26 Ref 6: Small Angle Approximation

In Chapter 3, Distances and Dead Reckoning, you saw a literal ‘rule of thumb’
that states that an object 100 ft. tall has an angular width of 1o at a distance of one mile.
This is an example of something called the small angle approximation. The rule of
thumb in principle allows you to estimate either the range to a distant object or, if you
know the range, estimate its size. There is more to it than just this, however. First, it is
important to know the range of validity. It’s only useful for angles less than about 15o.

The approximation can be understood a bit better when we use a different kind of
units to express angles: radians. There are 2π radians in a circle. This angular measure
makes it easy to express the length of an arc of a circle. If a circle is of radius r, and the
arc subtends and angle θ , then the length of the arc is rθ . The circumference of a circle
is 2πr , as should be well known. €

€ €
€

arc$=$r

r

Figure 1 Relation between angle in radians and the length of an arc.


  The conversion between degrees and radians is straightforward. There are 180o

in π radians. So, to go from angles to radians, you multiply the angle in degrees by

π =0.0175, or there are 0.017 radians in one degree. To go from radians to degrees,
180o
€ multiply the angle in radians by 180o =57.3, or there are 57.3 degrees in one radian.
π

€ Let’s take a situation where there’s a large angle and you know the height of a

lighthouse in the distance (h) and you measure its angular height, θ . You then want to
know the distance to th€e lighthouse, d. This is shown in Figure 2. The question is ‘how
do I solve for d, given h and θ ? You may want to look at the trigonometry refresher
video on the course website, but the ratio of the opposite to the adjacent sides of a right
triangle is the tangent of the angle. That is to say, €

€ h = tanθ
We can rearrange this to find d: d

€


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d = h
tanθ

€

h

d

Figure 2 A lighthouse of height h subtends an angle at some distance. We wish to find the distance.

Now, I haven’t told you whether we measured the angle in radians or degrees, but
in either case, we could find a table of tangents or use a calculator to find the tangent of
the angle, plug it into the above formula and we’d find our distance to the lighthouse.

For the primitive navigator, we don’t usually carry around calculators or tangent
tables. Certainly when I’m kayaking in the wind and waves, I don’t have the time to
mess around with tangent tables. The small angle approximation makes this kind of
calculation easier, where we can dispense with tables. Typically objects at a large
distance cover a small angle.

Figure 3 below shows what happens when angles get small. You can see the
height of the light house, h is part of the triangle ABC. Just beyond point C, you can see
the start of the arc of a circle. If the arc is part of a circle of radius r, then the triangle
ABC has r as a hypotenuse and d as a base. If the angle θ is measured in radians, then
the length of the arc is rθ .

€ € Arc(=(r
A
B
r

h

dC

Figure 3 Illustration of the small angle approximation.


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If we use our previous result, the length of the height, h can be found from the definition
of the tangent of an angle, which is the opposite over the hypotenuse:

h = d tanθ
If we knew r, we could also use the definition of the sine of an angle:

€ h = sinθ
r

To find the height:

€ h = r sinθ

Now, if you look at the figure, as the angle gets smaller and smaller, you should be able
to see that r and d almost become equal to each other. In addition, the length of the arc
of the circle almost becomes t€he same as the height. This implies that

h = r sinθ = d tanθ ≈ rθ ≈ dθ

Providing we measure θ in radians, rather than degrees. Here, the symbol “ ≈” means
“approximately equal to.”
In fact, for angles les€s than about 15o or 0.25 radians, this is a reasonable approximation
and is the sm€all angle approximation. Since r and d also become a€bout the same, a
reasonable approximation for the angles are:

sinθ ≈ tanθ ≈ θ
Where the angle is measured in radians.

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functions because they cannot be expressed as a finite algebraic expression, but rather

only by an infinite one. In the case of small angles, the first term in the polynomial series

for the sine and the tangent function is simply θ . The next term that contributes
1
anything for a sine and a tangent is about 3 θ 3. For angles less than 15o, this is a

correction of about 2%, so the small angle approximation is pretty good.
€

For a cosine function, the first two terms in the polynomial approximation are

cosθ ≈ 1 − 1 θ 2, where again, th€e angle is measured in radians. We could probably just
2
say that the cosine of a small angle is equal to one and be done with it for our purposes,

but I thought that I’d show the next term for completeness.

€ So, the main thing to remember is that if you have an angle in radians that’s less
than about 0.25, then you can directly use this to estimate distances or heights of distant
objects. Often times, the markings on rifle scopes and other instruments are graduated in

thousandths of a radian, called a milliradian or mil for short. When snipers need to make


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an accurate gauge of the distance of their target, they use the markings, called a reticle to
help gauge that distance.


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