What it shows
If you walk around any polygon and turn at each corner, the turns add up to one full turn of 360 degrees. From this the sum of the interior angles, (n minus 2) times 180 degrees, follows.
The science
At each corner the exterior angle is the angle through which you turn to face the next side. After walking around a closed polygon you face your original direction again, so the total of all the turns is a whole number of full turns, which is one turn or 360 degrees for any simple polygon. Each interior angle is 180 minus the exterior angle, so the interior angle sum is n times 180 minus 360, which is (n minus 2) times 180.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Chalk or a long rope on the playground
- A measuring tape or a pace stick
- A protractor made from card or a paper plate
- Cone markers or stones
- A notebook
- A friend
Safety
For this project: Do the activity in an open, clear ground away from traffic and in shade or in the morning.
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.
Step 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 or lay a rope in a large triangle on the ground and place a marker at each corner.
Stand at the first corner and face along the first side, then walk along it to the second corner.
At the corner turn to face the next side, and measure the turn angle with the paper protractor held on the ground. Note it down as the exterior angle.
Continue around the shape, writing each turn, until you are back at the start and facing your first direction.
Add up all the turns and write the sum. Repeat with a quadrilateral, a pentagon and an irregular six-sided shape.
Compute the interior angle at each corner as 180 minus the exterior angle, and add them.
Make a table with n, the exterior angle sum, the interior angle sum and the formula (n minus 2) times 180.
Check that for a regular polygon each exterior angle is 360 divided by n, and test with a regular hexagon whose turns are 60 degrees.
What you should see
Every shape gives a total turn of 360 degrees, within a few degrees of measurement error, and interior sums of 180, 360, 540 and 720 for the triangle, quadrilateral, pentagon and hexagon.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What is an exterior angle?
The angle between a side extended and the next side, the angle you turn at the corner.
02What is the sum of the exterior angles of any polygon?
360 degrees.
03What is the sum of the interior angles of a hexagon?
720 degrees.
04What is each exterior angle of a regular octagon?
45 degrees.
Take it further
Program a turtle in Scratch to draw regular polygons by turning 360 divided by n at each corner.
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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