Measuring Distant Objects

17 March 2014

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How can you measure the length of an object? Ideally, you use aruler, a tape measure, or some other measuring device.

But what if you can’t get to the object to physically measure it? Forinstance, what if it is the height of a mountain?

This week we will explore how certain objects can be measured.Today we will focus on terrestrial objects, such as pyramids orbuildings. Later this week we will look at how the distance to theearth and moon, and the size of the earth, was approximated over2000 years ago.

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Thales of Miletus

Thales, born around 624 B.C., was apre-Socratic Greek philosopher. Many,including Aristotle, regard him as thefirst philosopher in the Greek tradition.Thales’ rejection of mythologicalexplanations became an essential ideafor the scientific revolution. He was alsothe first to define general principles andset forth hypotheses, and as a result hasbeen dubbed the “Father of Science.”

One of the mathematical problemsThales solved was how to measure theheights of the Egyptian pyramids.

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Great Pyramid of Cheops

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Clicker Question

Could we take a long rope to the top and measure how much of the ropeit takes to reach the bottom?

A Yes that should work.

B No it won’t give the right height.

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Answer

Unfortunately, that won’t work. It would give the length of a diagonal sideof the pyramid. That is longer than the height of the pyramid.

If we could cut a hole in the pyramid straight down and drop the ropeuntil it hits the ground, measuring the length of the rope would give us theheight.

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Illustration of a Variant of Thales’ Method

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What did Thales Do to Measure the Height?

Thales discovered a way tomeasure the height of thepyramids.

Thales reasoned that if his height was the same as the length of hisshadow, then the same should be true for the pyramid.

He waited till his height equaled his shadow, then measured theshadow of the pyramid. From this he knew the height of the pyramid.

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Thales’ Discovery as a Beginning of Mathematics

Auguste Comte: “In light of previousexperience we must acknowledge theimpossibility of determining, bydirect measurement, most of theheights and distances we should liketo know... In renouncing the hope, inalmost every case, of measuringgreat heights or distances directly,the human mind has had to attemptto determine them indirectly, and itis thus that philosophers were led toinvent mathematics.”

Auguste Comte19th century philosopher

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At the heart of Thales’ discovery is the notion of proportionality.Plutarch, a Greek historian, gives another version of Thales’measurement:

“The height of a pyramid is related to the length of its shadow just asthe height of any vertical object is related to the length of its shadowat the same time of day.”

This is more powerful than what Thales did. We’ll discuss the ideabehind Plutarch’s statement in some detail, and make it precise.

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Proportionality and Scaling

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What happens when you put an image in a copy machine and enlargeor shrink the image? The pictures on the previous page are the same,except that the right-hand picture is enlarged to 200% of theleft-hand picture.

By doubling the size of the picture, each length gets doubled. If KingKong was 2 inches tall in the first picture, he’d be 4 inches tall in thesecond picture.

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Clicker Question

If you double King Kong’s dimensions, do you think his weight doubles?

A Yes

B No

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Answer

His weight would increase much more than twice. One way to think aboutthis is to think of a cube of some material. If each side length doubles,then the volume increases by a factor of 8. The weight would increase by afactor of 8.

It turns out that this is the reason why King Kong and flies the size ofhumans are fictional. Bones aren’t strong enough to handle the increasedweight. But, that is a story for another time.

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Similar Triangles

If we take one triangleand enlarge or shrink it,as a copy machine woulddo, we get anothertriangle. These twotriangles are calledsimilar.

Each angle of the smalltriangle is equal to oneof the angles in the bigtriangle. The equalangles are marked withthe same color.

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When the angles of one triangle are equal to the angles of another,then one triangle is obtained from the other by magnifying (orshrinking), and the two are similar.

If we think of taking the smaller figure and magnifying it with a copymachine, then the scale factor represents how much we magnify. Forexample 200% corresponds to a scale factor of 2, and 150%corresponds to a scale factor of 1.5.

The scale factor says by what factor each length grows when goingfrom the smaller figure to the larger figure.

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In the following picture, the right-hand picture was made by takingthe left-hand picture and magnifying it by 200%. Note that the twoline segments have doubled length but the size of the angle is thesame. Scaling does not change angles.

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The length of the segment EG can be found by multiplying the lengthof AB times the scale factor. Similarly, the length of EF is the lengthof AC times the scale factor, and similarly for the third sides.

Written as equations, if s is the scale factor, then

EG = s · ABEF = s · ACGF = s · BC

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By dividing, we can write theseas

EG

AB= s

EF

AC= s

GF

BC= s

We can write this withoutreference to the scale factor as

EG

AB=

EF

AC=

GF

BC

This is a useful set of equationscoming from similar triangles.

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In words, this relationship can be described as: “If two triangles aresimilar, then corresponding sides are proportional, meaning that theratio of the lengths of corresponding sides is the same.”

Corresponding sides represent an original side and a scaled side. Forexample, AB and EG are corresponding sides.

Corresponding angles are those drawn in the same color above.

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Clicker Question

These two triangles are similar. How long is the unknown side?

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Answer

The length is 2 inches. It is the solution of the equation

???

1=

3

1.5= 2

Another way to answer this is to note that, due to the top sides, thescale factor is 3/1.5 = 2. Thus, we have to multiply each length ofthe left-hand triangle by 2 to get the corresponding length in theright-hand triangle. Therefore, the unknown length is 2 inches.

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Proportionality

The relationship between corresponding sides of similar triangles is anexample of a proportional relationship. Two variable quantities aresaid to be proportional if their ratio is a constant.

Another example is the ratio of height to arm length in any photo ofKing Kong. No matter how much we scale the picture, the ratio willbe the same. If, say we scale the picture by 200%, King Kong’sheight will double, but so will his arm length. So, the ratio of heightto arm length will remain the same.

Another example comes from circles. The ratioof circumference to diameter (twice the radius)is always constant. The ratio is the number π.

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Plutarch’s Variant of Thales’ Method

Thales understood similar triangles, and how similar triangles couldbe used to measure heights.

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In this picture the two triangles are similar because they have equalcorresponding angles.

The blue angles are equal because the sun’s rays are parallel.

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Because these triangles are similar, corresponding sides areproportional. This means

height of clock tower

height of stick=

length of clock tower shadow

length of stick shadow

In this equation, we know or can measure three things, the twoshadows and the height of the stick. We can then solve for the heightof the clock tower.

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An Example

Suppose the stick is 3 feet high, it casts a shadow of 2 feet, and theclock tower casts a shadow of 25 feet.

Our equation

height of clock tower

height of stick=

length of clock tower shadow

length of stick shadow

then becomesheight of the clock tower

3=

25

2

We can multiply by 3 to get

height of the clock tower = 3 · 25

2=

75

2= 37.5 feet

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The benefit of using similar triangles is that we don’t have to wait forour shadow (or that of the stick) to be the same length as our height,as Thales did. We can do this measurement at any point in time.

One drawback to this method is that we need to have a sunny day, sothat we can see shadows.

Another drawback is that we may not be able to measure a shadow.We need to have space around us in order to measure the shadow.This wouldn’t be feasible in many situations. For instance, wecouldn’t measure the height of Organ Peak in this way.

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Next Time

We will explore some methods that will overcome the problemsmentioned in the last slide. Each of the methods we consider will usesimilar triangles.

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