The sightline, and the horizontal it gets measured against

0:00Until now, every trigonometry question started with a right triangle already drawn for you. A person is standing in a field. There is a tower some way off, and they want its height. There is no triangle here. There is a person, a tower, and some grass. So the work is not computing a ratio. It is manufacturing the triangle. And that turns on one decision which is easy to walk straight past.

0:30An angle needs two rays. The sightline from the eye to the top is one of them. The second one has to be chosen, and everything else depends on the choice. I built a hundred and eighty of these scenes out of four points each, and measured every claim here on all of them. Five points, each one something you could put a finger on. E is where the feet are. A is the eye, directly above E.

0:59D is the foot of the tower, and C is its top, the point being aimed at. The sightline is the ray from A to C. Now the second ray. From A, draw a line that is dead level, running out towards the tower. That level ray hits the tower somewhere. Call that point B. The opening between the sightline and the level ray, at the eye, is the angle we are going to read.

1:27Notice where B landed. Not at the foot and not at the top, but partway up. B sits at exactly eye height, because the ray that found it was level and started at the eye. The level ray through the eye is not the only level line here. The ground is level too. So why not measure the angle up from the ground, at the feet? The two rays are parallel, so they look interchangeable. They are not.

1:57The angle at the feet has its vertex at E, and it opens against a ray running from E, not from A. Different ray, different angle, different number. I read the same sightline both ways in all hundred and eighty scenes. They agree in exactly thirty six of them, and those thirty six are precisely the scenes where the eye is on the ground. Eye on the ground, they always agree. Eye above it, they always differ. Nothing breaks either way.

2:30The instrument that reads this angle is held up to the eye, and the level line it compares against runs through the instrument. The angle at the feet is not a number anyone in that field can obtain. Now do something that feels like vandalism. Erase the tower, the grass, the person. Keep the five letters and the lines between them. What is left is a triangle sitting on a four sided figure.

3:01The triangle is A, B and C. The four sided figure is A, B, D and E. Same drawing. Nothing added, nothing moved. Only the scenery went. The scenery made the scene recognisable, but it carried no part of the argument. Draw the person as a speck against a tower a hundred times their height and the reasoning is untouched. The drawing does not have to be to scale. The letters carry it.

3:32The triangle is only useful if it is right angled. So where is the right angle, and why? It is at B, where the level ray meets the tower. The ray from A to B is level, because we drew it level. The tower stands plumb, because towers do. A level line and a plumb line cross at a right angle. That is the whole justification, and it is the single assumption in the construction.

4:01I measured it at B in all hundred and eighty scenes, with a dot product rather than a protractor. Right angle every time. Let the object lean. Same eye, same run, same top height, but the top is no longer over the foot. Five hundred and forty leaning scenes. Four hundred and thirty two lost the right angle, and lost the four sided figure with it. The other hundred and eight are the ones where the eye was on the ground, where that figure had already flattened to a segment.

4:38So the right angle is not decoration. It goes when the plumb line goes. So what do you go outside and find out? One, the ground run, paced out from your feet to the foot of the tower. Two, the angle at your eye. That is the instrument reading. Three, your own height, ground to eye. That one you measure indoors, once. The strong claim is that those three pin the answer down exactly.

5:07I grouped all hundred and eighty scenes by the three of them together. Every group held exactly one object height. Then I grouped by two of the three at a time. Every one of those three pairings produced a group holding more than one height. Drop any one and the answer is genuinely not determined. So this is not a list of three convenient things. It is exactly what the problem needs, with nothing spare.

5:39Here is where the careless answer gets made. The angle is at A, the run is the side touching it, and the side facing it is B C. So the tangent of the angle is B C over A B, and B C comes straight out. And then people write that down as the height of the tower. It is not. B C runs from B to C, and B is at eye level.

6:04So B C is the part of the tower above your eye, and only that. The tower has two storeys, and the triangle only ever knew about the upper one. The full height is C D, and C D is C B plus B D. The triangle's answer falls short of the true height by exactly the observer's own height. Not nearly. Exactly. And adding that height back gives the true height, every time.

6:33The upper storey is computed. The lower storey is measured, and it is you. Why should the lower storey be exactly your height? Look at the four sided figure. Two of its sides are level: A to B is the level ray, and E to D is the ground. Two are plumb: E up to A is you standing, and D up to B is the bottom of the tower.

6:59Two pairs of parallel sides meeting at right angles. That is a rectangle. I measured all four of its angles in all hundred and eighty scenes. In a rectangle, opposite sides are equal, and E A and D B are opposite sides. E A is your height. D B is the lower storey. I measured both, on every scene, rather than reading it off a picture. They match every time.

7:28And that is why the correction does not depend on how far out you stand or how tall the tower is. The rectangle sets it, and the rectangle's vertical side is you. A vertical tower. You stand fifteen metres from its foot and take the sighting at ground level. The angle you read is sixty degrees. The height above eye level is fifteen times the tangent of sixty, and that tangent is the square root of three.

8:00So the height is fifteen root three. Its square is six hundred and seventy five exactly, and as a decimal it is a shade under twenty six metres. Nothing was added at the end, and not because the correction was forgotten. The sighting was taken at ground level, so the eye height was nothing, so the lower storey has no height. The rule was not skipped. The correction happened to be zero.

8:29That is the difference between a rule with exceptions and a rule that is always applied. Three more in the same shape, all sighted at ground level. Thirty metres out at thirty degrees gives a height whose square is three hundred. A taut rope twenty metres long from the top of a pole to the ground, meeting it at thirty degrees, gives a pole of exactly ten metres. And a kite sixty metres up on a string at sixty degrees needs a string whose square is four thousand eight hundred.

9:04Now one with an observer who has a height. She is one and a half metres to the eye, standing twenty eight and a half metres from a chimney, and the angle at her eye is forty five degrees. The tangent of forty five is one, so the upper storey is twenty eight and a half metres, the same number. And then the lower storey. One and a half. The chimney is thirty metres, not twenty eight and a half.

9:32One and a half in thirty is a twentieth of the answer. Small. Not nothing. Now the argument I want to break: that this is fussiness which only matters for tall things. Keep the observer and shrink the scene. She sights a ceiling fitting from one and a half metres away, again at forty five degrees. The triangle gives one and a half metres above her eye. The fitting is at three metres.

10:01Her own height is half the answer. Across all hundred and eighty scenes the correction's share runs from nothing at all, when the eye is on the ground, up to five eighths. It is largest for short things, close up. It is never rounding error. Two confusions left, and both cost marks. First, the angle is not up at the top of the object. There is an angle at C, between the sightline coming down and the tower. It is not this one.

10:35Those two are the two acute angles of the same right triangle, with one right angle to share between them. The side facing the angle at the eye is the side touching the angle at the top, and the other way round. So the tangent of one is the reciprocal of the tangent of the other, and that held in all hundred and eighty scenes. The two were the same number in only six of them, which are the six at forty five degrees.

11:05Second, the distance to the tower and the distance to its top are different distances. The run is A B, along the level. The sightline is A C, and it climbs. The sightline is the longest side, so it is strictly longer than the run, in every scene. At sixty degrees it is exactly twice the run. You stand fifteen metres out and look along thirty metres of air. Of the hundred and eighty sightlines I measured, only thirteen had an exact length at all.

11:41The sightline goes from the eye to the target, and the angle is measured against a level ray through that same eye. Not through the feet, not up at the object. At the eye, because that is where the instrument is. That choice plants a right angle in the scene, where the level ray crosses the plumb line of the object. The price is that the triangle starts at eye level, so it delivers the upper storey only.

12:11The lower storey is a rectangle's opposite side, which is to say it is your own height, and it gets added back. Three field measurements: the run, the angle, and your height. Those three determine the answer, and no two of them do. The picture makes the scene recognisable. The five letters do the work.