Working With Right Triangles
to Determine Heights and Distances

By understanding the relationship of particular sides in a right triangle, along with some simple trigonometric functions and helpful formulas, we can easily solve for unknown angles, heights, or distances.

One of the earliest examples of a geometric application is that of Thales, the ancient Greek philosopher and mathematician, who lived from about 640 BC. to 546 BC.  In trying to determine the height of a pyramid, he placed an upright stick at the foot of the pyramid's shadow and discovered that the relation between the pyramid's height and shadow was the same ratio as that of the stick's height to its shadow.  In other words, if a stick 3 feet high were to cast a shadow 5 feet long, while at the same time a pyramid's shadow measured 750 feet, we can then determine the height of the pyramid.

The pyramid, as well as the stick, can be visualized in the framework of a right triangle.  And though viewing the stick this way is a much smaller scale than that of the pyramid, yet their tangent ratios are identical, which means that their angles are also of the same degree.  We can, therefore, learn from the smaller what we need to know for the larger.  For in determining the tangent of an angle, the side "opposite" to the angle is divided by the one "adjacent" to it.  And for now, we are thinking of the "opposite" as the height, and the "adjacent" as the length of the shadow.

For instance,  suppose that an angle has an "opposite" side of  3 and an "adjacent" side of 4.  Dividing the opposite by the adjacent will give us a ratio of 0.75.  For the formula for the tangent ratio of an angle is "TAN =  OPPOSITE / ADJACENT,"  and for our example that means 0.75 = 3 <divided by> 4.

What I want you to also see in this is how helpful this formula can be in finding unknowns.  For since "tan = opposite / adjacent," then "tan * adjacent = opposite" and "opposite  /  tan  = adjacent."  So with this formula, and its re-arranging, we can determine  three different unknowns.

Let us now try this with the pyramid and stick.

The stick stands 3 feet and casts a shadow of 5 feet.  We are thinking of its height as the "opposite" to the angle, and the length of the shadow as the "adjacent" to the angle; and since TAN = OPPOSITE / ADJACENT, then 3 / 5 = 0.6.  So "0.6" is the tangent of the angle.

Now if we only knew the angle and the adjacent -- and not the opposite -- we could determine the "opposite" by multiplying  the adjacent by the tangent; and in this case it would be  5 * 0.6, which would result in 3.  Or if we didn't know the adjacent, we could divide the opposite by the tangent, which would be 3 / 0.6 to come up with 5.

Though the stick is much shorter than the pyramid, they still have the same angle degree caused by the sun.  Their heights and shadows greatly differ in length, but their own ratio between their heights and shadow lengths are the same.  They both have the same tangent ratio.  So because of that, we can use the same tangent ratio of the stick and its shadow to determine the height of the pyramid.

How High is the Pyramid?

Since the pyramid has the same tangent ratio as the stick of 0.6, then

0.6 * 750 = 450

So the pyramid is 450 feet in height.

Now wasn't that much easier than having to awkwardly climb up a 450-foot structure with an enormously long tape measure?

If Thales had a calculator in his day, there is also another way that the pyramid's height could have been easily determined -- and without using the stick.   One would merely have to measure the length of the pyramid's shadow and obtain the angle of elevation from the foot of that shadow to the peak of the pyramid.  The angle would be 30.96 degrees.   By inputting that number into a calculator and pressing the TAN (tangent) button, we come up with 0.6.  Then we can plug the value into our formula: "tan = opposite / adjacent," but re-arrange to "tan * adjacent = opposite" to now determine the value of the "opposite."  Therefore, 0.6 * 750 = 450.

To prove that the tangent ratio remains the same for the pyramid and stick, think of any acute angle.  Now if you added several right angles in it that were an inch or so apart, they would each still be sharing the same angle A.  And as the right triangle increases in size as we move from one of those right angles to the next, still each one maintains the same tangent ratio between their opposite and adjacent.  For instance, if the "opposite" side were 3 and the "adjacent" side 4, the ratio would be 0.75; but if these sides were 6 and 8, respectively, we would still have a tangent ratio of 0.75.  And this continues wherever we would place a right angle on the same acute angle, whether the vertex  of the 90 degree angle would be 10 inches away from the vertex of angle A, or 12 inches away, or 14 inches away, etc.

Because the height of the pyramid was the same ratio to its shadow as the stick was to its, we could have also used this following formula for determining the pyramid's height

A        C
---  =  ---
B       D

Here we have plugged in all the values except the pyramid's height:
:
3           x
----  =   -----
5          750

We can now solve "x" in either of the following ways::

1) Divide the second denominator by the first denominator, and then multiply the result by the first numerator.

2)  Or,  divide the first numerator by the first denominator, and then multiply the result by the second denominator.

This can also be double-checked by cross-multiplying: Just think of it in the shape of an "X" and multiply the first numerator by the second denominator.  Then multiply the first denominator by the second numerator.  Both results should be the same.  In this case, 2,250.

Looking More Into the Right Triangle

We considered, above, an example of using a right triangle to determine an unknown height; and there are also more examples to follow.  But let us first take a brief look at a right triangle; and with only three sides, it shouldn't take us too long to learn them.  These sides are called the hypotenuse (which is always the longest side), the adjacent (to the angle), and the opposite (to the angle).

The right angle is directly across from the hypotenuse, and it is sometimes labeled with a capital “C,” which would then correspond to the small “c” that labels the hypotenuse. The small square box, which is sometimes placed in the inside corner of the right angle, points to the hypotenuse.

In working with the same right triangle, the sides of the opposite and the adjacent will switch places when we move the focus from angle A to angle B, but the hypotenuse will still remain the same for either angle.

If the vertex of the right angle is in the bottom right of the right triangle and the vertex of angle A is in the bottom left, then angle B is at the top. The side across from angle A is called the “opposite” and usually labeled “a,” and the side across from angle B is usually labeled “b” and called the “adjacent” to angle A (but is also the  “opposite” to angle B).

The term “adjacent” signifies that that side is adjoining the angle. This is why side b is the adjacent to angle A, while side a is the adjacent to angle B.

The Pythagorean Theorem

There is a helpful formula called the Pythagorean Theorem that helps us in figuring out the lengths of a right triangle's sides. First, and moving clockwise with our above described right triangle, let us label the sides as “c” for the hypotenuse, “a” for the opposite, and “b” for the adjacent. The Pythagorean Theorem states that “c^2 = a^2 + b^2.” To see this demonstrated, let us suppose that c = 5, a = 4, and b = 3. By plugging these values into the formula, we then have c^2 = 25, a^2 = 16, and b^2 = 9. So squaring the lengths of sides a and b and then adding them will total 25, the same as the square of c.

We can also see in this that by knowing any two of the sides, we could then easily determine an unknown side. We merely need to re-arrange the formula, such as “c^2 - a^2 = b^2,” if we were wanting to determine the length of b. By plugging in our above values for c and a, we then have 25 - 16 = b^2. So, b^2 = 9.

We then need to find the square root of that answer to show the actual length. The square root of 9 is 3; so there you have it.

Using a calculator that has squaring and square root functions makes this very easy.

To make this even easier, if you have an HP calculator with a “solver” feature, try inputting this short program:

The calculator recognizes the “SQ” (square) function and displays the labels of HYPOT, OPP, and ADJ on the screen. With it, you can plug in any two values to determine the third with the press of a button. Just use the actual lengths of whatever two sides you want to input, and then press the button for the one you want to determine. The inputted numbers, as well as the result, will all be displayed in actual lengths, rather than in those numbers squared.

Another way you can determine side lengths – and by initially knowing the length of only one side – is by using a trig function with the angle and the known side. This is where we get into the “Soh-Cah-Toa,” which is a helpful mnemonic for remembering which trig function to use with which sides of the right triangle.

SOH

“Soh,” for instance, stands for “SINE(of the angle) = OPPOSITE / HYPOTENUSE.” “Sine” is often abbreviated as “sin,” but still pronounced to rhyme with “twine” even then. This formula means that the opposite divided by the hypotenuse will be the SINE of the angle. So if you inputted the angle into a calculator and pressed the SINE button, it would give you the same answer as when dividing the “opposite” by the “hypotenuse.”

For instance, suppose we have a right triangle with an angle of 45 degrees, its “opposite” is 9 inches, and the “hypotenuse” is 12.73 (rounded to the nearest hundredth). If we would input 45 into a calculator and press the “SINE” button, the result would be 0.707 (rounded to the nearest thousandth). But also if we divide the opposite by the hypotenuse (9 <divided by> 12.73), we'll come up with the same answer, 0.707.

Now let us see a practical way we can use this:

Figuring Wire-Lengths for an Antennae

Two feet down from the top of a 15-foot antennae in a large field, you want to run four guy wires at a 45 degree angle.  How long does each wire need to be?

First of all, try to visualize the problem in the framework of a right triangle. And be aware of what two sides you'll be giving your attention to. One of the sides will be the side you are trying to determine the length of, and the other side will be the one needful to solve that.

In our particular example, we have a 15-foot antennae, but the wires will attach at the 13-foot mark; so we can think of the antennae itself (up to the 13-foot mark) as being the side of the right triangle that is “opposite” the angle. So the “opposite” is representing the height. At the base of the “opposite” is the vertex for the right angle, and running along the bottom from it is the “adjacent” (which we are not given the length of for this problem, but we won't need to know that). The adjacent continues to the vertex of our 45 degree angle; and the line that angles up from that to where the wires will be connected is the hypotenuse (which, therefore, represents the length of one of the wires).

So since this problem will concern itself with the “opposite” and the “hypotenuse,” which of the formulas in “Soh-Cah-Toa” will we need for that? “Soh” is the only one that deals with both of these sides; so we'll be using the SINE (sin) function – and not the COSINE (cos) nor the TANGENT (tan) for this problem.

Remember, the “Soh” formula means “SINE(of the angle) = opposite <divided by> hypotenuse.”

But since we don't know the length of the hypotenuse, how could we re-arrange this formula, and what operation would we use, to determine it?

If you have trouble figuring that part out, try comparing it to the following simple equation:

3 = 6 / 2

In this formula, the “2” is in the same location as the hypotenuse; so how could we re-arrange the 3 and the 6, and what operation would we use, to come up with 2 for the answer? That's pretty simple:

6 / 3 = 2

So we can then re-arrange our formula the same way:

Hypotenuse = Opposite / Sin(angle)

But better still, you might also find this helpful.  What observation can you make from the following that will be helpful for you in remembering the re-arrangement and operation to use for a formula?

S=O/H               C=A/H            T=O/A

O=S*H              A=C*H           O=T*A
H=O/S               H=A/C            A=O/T

First of all, whether we are dealing with SOH, CAH, or TOA, if we want to determine the value of the second letter in any of these formulas, we simply swap places with that and the first letter and change the operation to multiplication.  We can also see that it is only when determining the value for the second letter that we will multiply  We also note that if we want to determine the value of the third letter in SOH, CAH, or TOA, we also swap that with the first letter and let the operation of division remain.  So that's all there is to it!

So in re-arranging our “Soh” formula to solve for the hypotenuse, we then have...

hypotenuse = opposite / sin(angle)

hypotenuse = 13 / sin(45)

hypotenuse = 13 / 0.707106781187

hypotenuse = 18.38

Therefore, four times that (for 4 wires) will give you 73.52 feet. Plus, in addition to that, you'll want to include whatever extra inches you'll need for tying the wires.

Here's another problem:

How High Up on the Building is the Ladder Leaning?

If a 20-foot ladder, at an angle of 60 degrees, is leaning against a building, how high up on the building is the tip of the ladder?

In putting this problem into the framework of a right triangle, we have the 20-foot ladder represented by the hypotenuse; and the unknown height up the building represented by the “opposite.” So this will also use the “Soh” formula; and, therefore, the “SINE” function.

sin(angle) = opposite / hypotenuse

In our previous problem, we used this formula to solve for the unknown hypotenuse. Now, however, we will be solving for the unknown “opposite.”

sin(60) = opposite / 20

Therefore,

sin(60) * 20 = opposite

0.866025403784 * 20 = opposite

opposite = 17.32

So a 20-foot ladder, leaning at a 60-degree angle, will reach up to 17.32 feet on the side of a building.

Triangles with 30-60-90 and 45-45-90 Degree Angles

The 30-60-90 Degree Triangle

In the above examples, we used one right triangle that had a 60-degree angle, and another with a 45-degree angle.  These triangles have ratios that can be easy to remember.  For in a 30-60-90 degree triangle, the hypotenuse will always be twice the length of the shortest side; and the middle-length side will be "(the square root of 3) times the length of the smallest side."  So if the hypotenuse is 10, what would the other sides be?  The shortest side would be 5; and, using the square root function of a calculator, the middle-length side would be  8.66025403785.  So if we know the length of just one of the sides, we can then easily figure out the length for the others.

Example in Using the 30-60-90 Degree Triangle

So let's try this.  Suppose, for instance, that a kite is being flown, and the 300-foot string is at a 60-degree angle.  How high is the kite?

We can think of the string as being represented by the hypotenuse of a right triangle, and the side "opposite" from the angle as representing the height, which we want to determine.  That height we can visualize as an imaginary line coming straight down from the kite to the earth, and then over at a right angle to where the kite-flyer is standing.  The distance from the vertex of that right angle to the kite-flyer is represented by the adjacent side of the angle.

Now since the hypotenuse is going to be twice the length of the shortest side, and the middle-length side will be "(the square root of 3) times the smallest side," we can come up with the following:

length of string (hypotenuse) = 300 feet
distance from vertex of right angle to kite-flyer = 150 feet
height of kite = 259.81 feet (rounded to the nearest hundredth)

You can also double-check that by multiplying the adjacent (150) with the tangent of  angle A.  (The tangent of 60 is 1.73205080757.)   The result will be the same.

Suppose, though, that the string had been 200 feet long, and at the same angle.  How high would the kite be then?

length of string (hypotenuse) = 200 feet
distance from vertex of right angle to kite-flyer = 100
height of kite =  173.21 feet (rounded to the nearest hundredth)

The 45-45-90 Degree Triangle

For this particular triangle, you need to only remember that the hypotenuse is going to be "(the square root of 2) times either of the "legs" of the right angle (for they are both the same length).  So to find the length of the other sides, simply divide the length of the hypotenuse by the square root of 2.  For example, if the hypotenuse is 12, then each of the other sides will be 8.48528137426.

And so, if you didn't know the length of the hypotenuse, but did of the other sides, multiplying that length by the square root of 2 would give you the hypotenuse.  If, for instance, the adjacent and opposite sides were each 8, then the hypotenuse would be 11.313708499.

And since the opposite and the adjacent sides are the same length, then if we were to walk out from the base of a building until we could see its top at a 45 degree angle, the height of the building would be the same distance we walked.

Or if we didn't know the length of a long beam that is leaning against a structure at a 45 degree angle, but we did know that the base of that beam was 12 feet out from the building, or that it reached 12 feet up to the side of it, then we could easily figure that the beam would be 16.97 feet long.  For we merely need to multiply 12 by the square root of 2 for that answer,. which can be done very easily with a calculator.

Determining an Angle

Here's one more problem for the “Soh” formula that deals with determining the ratio of an angle.  The hypotenuse is 15, and the opposite is 12. What is the degree of the angle?

Remember: “Sin(angle) = Opposite / Hypotenuse.”

So 12 / 15 = 0.8

If you then input that number (0.8) into a calculator and press the arcsine button, it will give you “53.13” as the angle degree.

CAH

Figuring More Antennae Wires
(with a different formula)

If in a field, you had an antennae and didn't know the height of it, but you wanted to place 4 guy wires that would be grounded 10 feet out from around its base and that would have an angle of elevation of  50 degrees toward the antennae, how long would the wires have to be?

Again, let us put this in the framework of a right triangle. This time, however, we don't know the height (which would be represented by the “opposite” side of the right triangle), but we do know the distance from the base of the antennae to where the wires will be grounded. That distance is represented by the “adjacent” side of the right triangle. And we also know the angle of elevation that the wires are to have, but we don't know how long those wires need to be. How could we find the solution for this?

Since we don't know the wire-length (represented by the hypotenuse) nor the height of the antennae (represented by the “opposite”), then we can't use the “Soh” formula. But in looking at the “Soh-Cah-Toa” mnemonic, we see that “Cah” will use in its formula both the “adjacent” (which is representing the distance from the base of the antennae to where the wire is to be grounded) and the “hypotenuse.”

“Cos” is an abbreviation for “cosine,” another helpful trigonometric function.

We can now input what we know into this formula:

(cos)50 = 10 / hypotenuse

Since we need to determine the hypotenuse, we will have to re-arrange this formula and choose the right operation; and since “cos(angle) = adjacent / hypotenuse,” then “adjacent / cos(angle) = hypotenuse”:

10 / cos(50) = hypotenuse

10 /  0.642787609687 = hypotenuse

hypotenuse = 15.56 (rounded to the nearest hundredth)

We will, therefore, need 62.24 feet of wire (for the total of 4), plus whatever extra inches needed for fastening them.

How High Up the Antennae Were the Wires Attached?

By the way, as you probably figured, we did not need to know the height of the antennae for the solution because our only concern was that the wires would be grounded 10 feet out from the base of the antennae and then have an angle of elevation of 50 degrees to the antennae. But can you figure out how high up on the antennae these wires were attached? Which of the “Soh-Cah-Toa” formulas will you need to use for that?

With the information we now have for the hypotenuse (15.56), the adjacent (10), and the angle (50), we can use either the “Soh” formula or the “Toa” in order to find the height to where the wires fastened. For since they both involve the “opposite” (which is representing that height), we have the choice of using either that one that also includes the hypotenuse or the one that includes the adjacent.

If we chose the “Soh” formula, so that we can determine the “opposite” by inputting the value of the hypotenuse, the formula will look like this:

sin(50) = opposite / 15.56

We then re-arrange this formula to determine the value of the “opposite”; and since “sin(angle) = opposite / hypotenuse” then “sin(angle) * hypotenuse = opposite.”

sin(50) * 15.56 = opposite

0.766044443119 * 15.56 = opposite

opposite = 11.92
(for how many feet up the antennae the wires are attached,
and rounded to the nearest hundredth)

If, however, we used the “Toa” formula, so that we could determine the “opposite” by inputting the value of the adjacent, it could look something like this:

tan(50) = opposite / 10

1.19175359259 = opposite / 10

Then we re-arrange the formula to the following:

1.19175359259 * 10 = opposite

opposite = 11.92 (rounded to the nearest hundredth)

How Far is the Base of the Ladder from the Building?

Here's another: If a 25-foot ladder were leaning on a 65-degree angle against a building, how far would the base of it be from that building?

When we put this in the framework of a right triangle, the leaning ladder is represented by the hypotenuse, and we know the angle; but we need to find out the distance the base of the ladder is from the building, and this distance will be represented by the adjacent. So, again, we will need the formula that will deal with the hypotenuse and the adjacent; and that is “Cah.”

Since this above formula is so, then the following also is:

So the base of the ladder is 10.56 feet away from the building.

And if we divide the adjacent by the hypotenuse (10.56 / 25), we will come up with “0.422,” which is the same as inputting 65 (our angle) into the calculator and pressing the COS button. And inputting 0.422 into the calculator and pressing the arccosine button will give you the angle of 65 degrees.

TAN

How Tall is the Lighthouse?

Imagine that you were visiting a lighthouse and wondered about its height. You took 12 steps out from the base of it, and figured that to be about 36 feet. With an eye-level of 5.5 feet, you looked up at an angle of 70 degrees to the very top of it. How tall is the lighthouse?

As we put this problem in the framework of a right triangle, we can think of the “opposite” side of the right triangle running up the height of the lighthouse, and with the vertex of the 90-degree angle 5.5 feet up from its base (at our eye-level). The 36 feet that were walked from the base of the lighthouse is represented by the “adjacent” side of the right triangle. We also know that the angle from our eye-level to the top of the building is 70 degrees. However, we don't know the length of the hypotenuse, but that is not necessary for this problem.

So with this information, which of the three formulas will we need from “Soh-Cah-Toa”?

Since we know the adjacent and want to solve for the opposite, then we will use the “Toa” formula:

Tan is an abbreviation for tangent, another trig function.

By plugging in the information we have so far, we come up with this:

tan(70) = opposite / 36

We then re-arrange the formula to solve for the opposite:

tan(70) x 36 = opposite

2.74747741945 x 36 = opposite

opposite = 98.91 (rounded to the nearest hundredth)

Since the vertex of angle A is at our eye level, then we need to add that additional height of 5.5 feet to our answer.
So the actual height of the lighthouse would be 104.41 feet (rounded to the nearest hundredth).

How Far Away is that Building?

How about this one? You are walking down a very level road. In the not-too-far-away distance, you can see a building that you know to have a height of 100 feet. If from where you are, you can see at a 20-degree angle to the top of the building, how far are you from the building?

When thinking in terms of a right triangle, we'll let the building's height be represented by the “opposite” of the right triangle, and the distance we are wanting to determine will be the “adjacent.” So with that in mind, it is the “Toa” formula that we are again going to use; but with this problem, we will be solving for the “adjacent” rather than the “opposite.”

If “tan(angle) = opposite / adjacent,” then...

adjacent = “274.75” (rounded to nearest hundredth)

So the distance to the building is 274.75 feet.

How Far is it Across the River?

Here's another. Suppose you were walking along a peaceful river, and you began to wonder about the width of it. If you could see on the opposite bank of that river a large statue, how could you determine the distance across? Which of the “Soh-Cah-Toa” formulas would be the one to use for this?

To figure this, as you stand directly across from the statue, consider where you are as being the vertex of the right angle, and the distance from that point to the statue to be the “opposite” of the right triangle. Now walk about 50 feet to form the “adjacent” part of the right triangle, and be sure to keep on a straight course that won't mess up the right angle. Now, where you stop will be the vertex of angle A. From there, figure out the angle it is to the statue. That line of sight will be the hypotenuse, but you don't need to know its length, only the degree of angle A to the statue. For this example, we will say that the angle is 83 degrees.
So you will once again need the “Toa” formula for this, since you know the “adjacent” (50 feet), but want to determine the “opposite” (distance across the river).

NOTE: Prior, we thought of the “opposite” as representing the height; but now we have “placed the right triangle on its side” in order to determine not the height, but the distance – or, in other words, not the vertical, but the horizontal.

tan(83) = opposite / 50

8.14434642797 = opposite / 50

We then need to re-arrange our formula to solve for the “opposite.”

Since the above formula is so, then the following is also true:

8.14434642797 x 50 = opposite

opposite = 407.22 (rounded to the nearest hundredth)

So the river is 407.22 feet wide in that area.

As we have seen, trig functions can be very helpful; and remembering the three sides of a right triangle, along with what “Soh-Cah-Toa” stands for, will help us in that.

SOH
Sin(angle) = opposite / hypotenuse

CAH

TOA

I also included these formulas in the “solver” section of an HP calculator, as three separate formulas. They look like this:

SOH:SIN(A)=OPP/HYPOT

The “SOH:,” "CAH:," and "TOA:" are not necessary to run these.  They are merely titles, but they help me remember the sides these certain trig functions work with.  The calculator recognizes SIN, COS, and TAN as trig functions. When I run the one for "SOH," for example, the labels “A,” “OPP,” and “HYPOT” will be displayed in the screen. So by inputting values for any two of those displayed will allow me to solve for the unknown at the press of a button, and I don't have to do do any re-arranging of the formula nor select an operation.  So it works very well.  NOTE: The “/” is not actually used, but my keyboard doesn't show the division sign that the calculator uses – a dash with a dot above and below it -- so I'm using this as a substitute.

How High is the Kite?

Here's a final one: When I was about 7 years old, a teenage neighbor was flying a kite and kept adding more string to it. There were several of us kids there, and he would keep sending us down to the store to buy more string -- and even some sandwiches later on. (This is a true story.) Eventually, the kite was so high, we could no longer see it. When he finally started reeling it in, he used a broom handle to wrap the string around -- and it took him several hours. While it was at its highest point, how could we have determined the actual height of the kite?

Suppose that all the balls of string tied together equaled 3,000 feet, and with an angle of elevation of 61 degrees. What would be the actual height of the kite?

As we think of this in terms of a right triangle, we can picture the 3,000 feet of string as the length of the hypotenuse. Now we just need to figure the height, which will be the side of the right triangle that is opposite the angle. Which of the formulas in Soh-Cah-Toa do we need that deals with the hypotenuse and opposite? We need the “Soh.”

sin(angle) = opposite / hypotenuse

sin(61) = opposite / 3,000

0.874619707139 = opposite / 3,000

Now we re-arrange the formula to determine the opposite:

0.874619707139 * 3,000 = opposite

opposite = 2,623.86 (rounded to the nearest hundredth)

The kite at its highest elevation was at 2,623.86 feet – just 16 feet and 2 inches short from being a half a mile!

-- Tom Edwards (tedwards@onemain.com)