What it shows
You can find a distance you cannot measure directly by measuring a baseline and two angles. The sine rule in the triangle then gives the width.
The science
In any triangle with sides a, b, c opposite angles A, B, C, the sine rule says a / sin A = b / sin B = c / sin C. If a baseline AB has length d and the angles at A and B to a landmark P across the water are known, the angle at P is 180 minus A minus B. The distance AP is then d x sin B / sin P. The perpendicular width is AP multiplied by sin A. Errors come mainly from the angle readings, so a baseline that is long enough to make the angles near 60 degrees gives better results.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A measuring tape or a 10 m rope marked in metres
- A protractor and a plumb line, or a clinometer app on a phone
- Two sticks or flags for the ends of the baseline
- A flat piece of cardboard 30 cm by 30 cm for a sighting board
- A straw taped along its edge for sighting
- A pencil and graph paper
- A calculator with sin
Safety
For this project: Keep well away from the water edge. An adult must come along and stay with the group. Do not climb trees or walls to take readings.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Choose a landmark P on the far side, such as a tree or a post, and a straight dry stretch along your own bank.
Fix a stick at point A, walk along the bank 20 steps or about 15 m and fix a second stick at B. Measure AB with the tape.
Stand at A and sight along the straw to P, then to B. Read the angle between the two lines with the board and protractor. Call it angle A.
Repeat the same at B to find angle B, sighting to P and to A.
Calculate angle P = 180 minus A minus B and draw the triangle to scale on graph paper to see if it looks right.
Use the sine rule to find AP = AB x sin B / sin P. Then find the width as AP x sin A.
Repeat with a longer baseline and compare the two answers. A common mistake is to put the calculator in radians, so check it is in degrees.
Check the result by pacing the width at a place where a bridge or a narrow part lets you measure directly, and note the percentage error.
What you should see
An estimate of the width with a stated error, for example within 5 to 10 per cent of the measured value, and a scale drawing that matches the calculation.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1State the sine rule.
In a triangle a / sin A = b / sin B = c / sin C.
2Why do we need a baseline?
We need one known length, and then angles are enough to find the rest of the triangle.
3What happens if the baseline is very short?
The angles A and B nearly add to 180 and small angle errors cause large mistakes.
4How is this used in real life?
Surveyors use triangulation to measure land, and the Great Trigonometrical Survey of India used it to map the country.
Take it further
Use two landmarks to measure the distance between them across the water using two baselines.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
Free ebooks for this project
Science projects
Maths Working Models and Activities
Make maths visible: models for number, geometry, algebra and data
Science projects
Science Projects for Class 9 and 10
Models and experiments for high-school physics, chemistry and biology