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# Challenge: Make a Scale Map of the Island's Coastline

Updated Monday, 28th January 2008

The science behind making a scale map, part of the BBC/OU's programme website for Rough Science 2.

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Surrounded by miles of inaccessible bays, sandy and rocky beaches and an abundance of wildlife and tropical vegetation, including thick, forbidding mangroves, mapping the island would be a challenge for professional surveyors armed with the latest equipment.

But with only very basic tools - and not even a ruler - would we be able to produce a map of the Northern part of Carriacou in just three days? Mapping involves measuring angles and distances. Without any rulers, we first needed to define some unit of measurement for length. We could have just used some arbitrary unit, such as an arm length, as a standard distance. But then we would have had to give all the distances in terms of this length, so a mountain might be 2000 'arm lengths' away. We decided we'd try to get an estimate for a metre instead.

How can we measure a metre?

We relied on the fact that most people know their own height to within a centimetre. So we measured each of the Rough Scientists in turn and from these figures deduced what a metre length would be. Height line-up

Now that we have a metre measurement, what can we use to make the mapping of distances easier?

We used two measuring devices: a wheel and a length of string. The circumference of a wheel from a wheelbarrow was approximately 1m and by pushing it forward and counting each turn of the wheel, we could make metre measurements. We used a length of string with knots every 10m as a second check. Because some of the island's Northern coastline is inaccessible, we needed another way to measure the parts we couldn't visit. This is where trigonometry comes in.

Firstly, you mark out a baseline on ground that is as flat as possible. We needed a method of measuring angles and so we made a giant protractor. The angles were marked on it using a folded circle of paper and a 10mm spanner as a guide.  Using the wheel, string and protractor - and trigonometry - we would be able to work out both the heights of landmarks and their distance away.

If you place the protractor horizontally and put it in line with the baseline, then hold your eye up to the central point, you can read off the angle between the baseline and the point you're interested in. You can also use it to measure vertical angles if you hold it vertically and use a bob.

How do we work out the distance to an object?

Trigonometry is based around right angled triangles. So, to work out the distance to the object, we need to first make two right angled triangles.

Draw a straight line on the ground 200m long. Stand at each end of the marked length and measure angles A and B.You will now have two triangles with the distance to the object being a common side to both triangles (this is D). Now one way to find out the distance to the object at right angles to the 200m line, distance D, is to find C1 and C2.

We use the formula the tangent of the angle (TAN) = LENGTH OF THE OPPOSITE SIDE / LENGTH OF THE ADJACENT SIDE.
TAN A = D / C1 therefore D = C1 x TAN A

To work out C1, we can replace C2 in the equation with (200-C1).

These two equations can be re-written as:D = C1 x TAN A and D = (200 - C1) x TAN B which is the equivalent to D = (200 x TAN B) - (C1 x TAN B)
The two equations can be combined to read:C1 (TAN A + TAN B) = 200 x TAN B

To now find C1 we divide both halves of the equation by (TAN A + TAN B):C1= 200 x TAN B / (TAN A + TAN B)
And from that: D= C1 x TAN A

If you know how to work out the distance to the object, you can now work out its height. For example, if you measure the vertical angle of the object to be 35o and you have already calculated that it is 1000m away, then: tan(35o) = opposite/adjacent = (height)/1000 height = (1000 x tan 35o)

Measuring the height of an object using trigonometry.

Using this equation, we can work out the height of these objects if we have the angle.

ANGLE = 45o TAN 45o = 1Therefore: HEIGHT = (1000x1) =1000m ANGLE = 70o TAN 70o = 2.747

Therefore: HEIGHT = (1000 x 2.747) = 2747m ANGLE = 10o TAN 10o = 0.176Therefore: HEIGHT = (1000 x 0.176) = 176m

Using these techniques, it was possible to map the Northern part of the island.

We drew baselines on the beach of each bay we could get to, then estimated how far away the extreme points of the bay were by measuring the angles between the headlands and their baseline. We also took a compass bearing and drew a rough sketch. The compass was made from a magnetized needle stuck into two pieces of cork. When it was floated in a cup of water, it pointed to magnetic North.

Back at the lime factory we had to piece together the shapes, distances and bearings of all the bays. Because the instruments we made were rudimentary and calculations were at times rough, the map had limited accuracy. All things considered though, it's amazing what you can do with a few bits of wood, string and mathematics!

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Mathematics of Cartography - from the Department of Mathematics, Rice University website

Mathematics - links to maths sites on the Berrien County Intermediate School District site

## Books

Advanced Physics by Tom Duncan, John Murray

Infinite Perspectives: Two Thousand Years of Three-Dimensional Mapmaking by Brian M. Ambroziak and Jeffrey R. Ambroziak, Princeton Architectural Press

Landmarks of Mapmaking by R. V. Tooley and Charles Bricker, Wordsworth Editions Ltd 1989 (currently out of print)