What it shows
Any triangle, copied and turned upside down, joins its twin to make a parallelogram with the same base and height. So the triangle has exactly half the area, which is why area = 1/2 times base times height.
The science
A parallelogram of base b and height h has area b times h, since it can be cut and slid into a rectangle. A diagonal splits it into two congruent triangles, so each triangle has an area of half of b times h. This works for acute, right and obtuse triangles as long as the height is measured at right angles to the base, even when it falls outside the triangle.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Coloured paper or old greeting card, 4 sheets
- Scissors
- A ruler and a pencil
- Glue
- Squared paper
- A plain chart sheet for the display
Safety
For this project: An adult should help younger children with scissors. Keep cut scraps away from small children.
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.
Draw an acute-angled triangle on coloured paper with a base of 10 cm and a height of 6 cm and cut it out carefully.
Trace around it on another sheet and cut out a second triangle that is exactly the same size.
Place the two triangles together along a side, with one turned upside down, to form a parallelogram. Stick them down on the chart.
Measure the base and the height of the parallelogram with a ruler and calculate its area. Note that height is the straight distance between the parallel sides.
Write that the area of one triangle is half of this value, so 1/2 times 10 times 6 is 30 square cm.
Repeat with a right-angled triangle, where the two copies form a rectangle, and with an obtuse triangle, where the height falls outside.
Check each answer by laying the triangle on squared paper and counting the squares, taking parts of squares together.
Write a table with the base, height and area for each of the three triangles on the chart.
What you should see
Each pair of triangles forms a parallelogram, and the squared paper count matches half base times height within a square or two for all three triangles.
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 the formula for the area of a triangle?
Half times base times height.
2Why is the height not the slanting side?
Height is measured at right angles to the base.
3Does the formula work for any triangle?
Yes, every triangle has a twin that makes a parallelogram with it.
4What is the area of a parallelogram?
Base times the perpendicular height.
Take it further
Show that triangles on the same base and between the same parallel lines have equal area by drawing three of them.
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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