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FINDING THE HEIGHT OF A TREE

If you want pinpoint accuracy, there are laser devices on the market that will give you precision in a split second. This is more for those who want to know if a tree will hit their roof when it blows down. Or just because the basics of trigonometry are interesting.

YOU WILL NEED

 A protractor

 A pencil

 A bit of Blu Tack

 A calculator

Trigonometry has to do with triangles. It’s the branch of mathematics that examines the relationships between the lengths of the sides and the angles. You probably already know that the internal angles of a triangle always add up to 180°. That is one of the first things you learn – so if we know one angle of a triangle is 40° and another is 80°, the last has to be 60°, because 40 + 80 + 60 = 180.

So whether it’s an equilateral triangle, where all the sides and angles are equal:

Or an isosceles triangle, where at least two sides are equal:

… the internal angles always add up to 180°. You might also be interested to know that angles along a straight line also add up to 180°. If you think about it, 180° is half a circle of 360°.

If you drew a straight line and crossed another through it, you would have two right angles of 90° – for a total of 180°. Four right angles, or 4 × 90 = 360 – the full turn.

That means, just as a matter of interest, that if you know any internal angle of a triangle, you can extend a straight line and also know the external angle. That might come in handy one day.

Sadly, there isn’t space enough here to cover all the interesting aspects of triangles. We’ll concentrate on one very specific task – finding the height of something using trigonometry – a word that means ‘triangle measuring’. It might be a tree or a building. In theory, it could be a person, though this is more a method for big objects.

First, pace a distance from the base of your object. Use a little common sense and pace out a fair way – 30, 60 or 90 yards. Those are not accidental choices. One problem with metres is that they have no physical reality, whereas a yard is a man’s pace. It’s possible to pace out a field in yards, for example, but not metres. However, a metre is – as near as makes no odds for our purposes – 3ft 4in. That means that 10ft (3 × 3ft 4in) is very close to 3m. So we chose a distance from the tree that could be expressed fairly easily in both yards and metres. Sixty yards is 180ft, or 18 × 3m = 54m.

You now have the base of your triangle. You are still missing the height of the tree, the hypotenuse (the longest side diagonal) and the angles. Where the tree touches the ground is 90°, which is what will make this work. The next part works for all right-angled triangles – triangles with a 90° angle in them.

Now, part of trigonometry lies in recognising the relationship between the lengths of the lines and the angles. If you lengthen a line in a triangle, the angles change. We express that relationship with the words ‘sine’, ‘cosine’ and ‘tangent’. On a calculator, they are usually written as Sin, Cos and Tan.

Each one of those three expresses a relationship between two of the triangle lines and an angle. The mnemonic for all three is SOH-CAH-TOA (pronounced so-car-toe-a). The letter ‘O’ is for Opposite, ‘H’ is for Hypotenuse and ‘A’ is for Adjacent – the side closest to the angle.

Sine is the relationship or ratio between the Opposite and the Hypotenuse. You find the sine of the angle x by Opposite/Hypotenuse. The most famous example is the 3–4–5 triangle. (Pythagoras used it to prove the relationship between the sides on a right-angled triangle was a² + b² = c², where c is the hypotenuse.) In this case that would be 3² (3 × 3) + 4² (4 × 4) = 5² (5 × 5).

In this example, if we wanted to find the angle x and had all three side lengths, we could use Sin, Cos or Tan to do it.

Sin = Opposite divided by Hypotenuse = 4/5 = 0.8

Cos = Adjacent divided by Hypotenuse = 3/5 = 0.6

Tan = Opposite divided by Adjacent = 4/3 = 1.33 recurring

It is possible to work out which angle would produce each of those ratios, but it’s pretty advanced. Before calculators, schoolboys used log books, where the answers had been worked out and could be checked or confirmed. Today, you’ll probably use the Sin⁻¹ button – inverse sine – to turn that ratio back into an angle.

Sin x = 0.8

x = inverse sin (Sin⁻¹) of 0.8 = 53°

Now that we’ve covered the basics of trig – back to our tree. We have the base of the triangle. However, we don’t know the height of the tree, nor the hypotenuse. We need to know an angle. For this, we use a protractor, a pencil and a blob of Blu Tack.

Lie on the grass and get as low as you can with the protractor. (We found we couldn’t lay it right on the ground because we couldn’t get an eye low enough to look along the pencil and see the top of the tree.) Holding it steady and just off the ground, raise the pencil until the tip points at the top of the tree. Read off the angle – in our example it was 50°.

We still couldn’t use sine or cosine (Sin or Cos) as we didn’t know the hypotenuse. However, we could use the Tan ratio to discover the missing height.

Tan 50° = Opposite (h for height) divided by Adjacent (60)

Tan of 50° is 1.19 (to two decimal places), which we can plug into the equation:

1.19 = h/60

To get h alone, we still have to do something about that ‘divided by 60’. You may know that an equation means two sides that are equal. You can double one side and, as long as you do the same to the other side, it’s still equal. So 2x = 4 is the same as 4x = 8.

If we multiply both sides by 60, that will make the ‘divided by 60’ disappear – and leave just h: the height of the tree.

1.19 × 60 = h

So 71.4 = h

That figure of 71.4 is in yards, of course. We’d multiply that by three to put it in feet – a mature Douglas fir that turned out to be 214ft tall.

Just to be clear, 60 yards is approximately 54m. (It’s actually 54.864m, so a little over.) To prove it works, we’ll use the metre figure here. If we plug that into the same equation, the answer comes out in metres.

Tan 50° = Opposite (h) divided by Adjacent (54.864)

1.19 = h/54.864

1.19 × 54.864 = h (in metres)

h = 65.288m

Or in feet: 1.19 × 180 = h

h = 214ft tall

Now, this wasn’t as precise as we’d have liked – that angle will always be a best estimate and it’s a key number. However, the basic idea – pace off 60 yards, estimate the angle and use Tan x = Opposite divided by Adjacent to find a height you can’t reach otherwise – is not beyond you.

Finally, you could estimate the key angle by kneeling and raising an outstretched arm. Remember to tell those around you that it’s not a Nazi salute. That could be really important.