What it shows
The point where the three medians of a triangle meet is its centroid, and a flat triangle balances there. Its coordinates are the average of the three vertex coordinates.
The science
A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid, which divides each median in the ratio 2 to 1 from the vertex. For vertices (x1, y1), (x2, y2) and (x3, y3) the centroid is at ((x1+x2+x3)/3, (y1+y2+y3)/3). A uniform flat triangle has its centre of mass at the centroid, so it balances on a point there.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A sheet of thick cardboard
- Graph paper to glue on it
- A pencil, ruler and scissors
- A sharp pencil tip or a blunt needle stuck in an eraser as a pivot
- Glue
- A compass
- Calculator
Safety
For this project: An adult must supervise cutting. Keep the needle covered and use a blunt point for balancing.
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.
Glue graph paper on cardboard and draw a triangle with vertices at (1, 1), (9, 2) and (4, 8) on the grid.
Cut the triangle out carefully along its sides.
Find the midpoint of each side using ((x1+x2)/2, (y1+y2)/2) and mark it.
Join each vertex to the midpoint of the opposite side with a ruler. The three lines should meet in one point.
Calculate the centroid using the average formula and check that it matches the point where the lines meet.
Measure along one median to check that the centroid is two thirds of the way from the vertex.
Balance the triangle on the tip of a blunt point at the centroid, and see it stay level. Then try a point a centimetre away.
Repeat with an irregular quadrilateral cut into two triangles and find its balance point by joining the two centroids, weighted by their areas.
What you should see
The three medians meet at a single point which the formula predicts, and the cardboard triangle balances there without tipping.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is a median?
A line from a vertex to the midpoint of the opposite side.
2What ratio does the centroid divide each median?
2 to 1, with the longer part next to the vertex.
3What is the centroid of vertices (1,1), (9,2), (4,8)?
((1+9+4)/3, (1+2+8)/3) = (4.67, 3.67).
4Why does the triangle balance there?
The centroid is the centre of mass of a uniform flat shape.
Take it further
Cut a cardboard map of India and find its balance point by hanging it from two points and drawing plumb lines.
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.
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