What it shows
On a circle, an arc makes an angle at the centre exactly twice the angle it makes at any point on the remaining circumference. Moving the pin around the circle shows that the angle at the edge does not change.
The science
In a circle, the angle subtended by an arc at the centre is twice the angle it subtends at any point of the remaining circumference. The proof uses isosceles triangles formed by the radii, since the exterior angle of a triangle equals the sum of the two opposite interior angles. A direct result is that angles in the same segment are equal, and the angle in a semicircle is 90 degrees.
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 thick cardboard or thermocol sheet 30 cm square
- A compass and a pencil
- About 12 pins or small nails with heads
- Coloured thread
- A protractor
- A white sheet to glue on the board
- A ruler
Safety
For this project: An adult must supervise the use of pins and compass points. Push pins in fully so heads are the only part left above, or cover points with clay.
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.
Fix the white sheet on the board and draw a circle with a radius of 10 cm using the compass. Mark the centre O with a pin.
Place two pins A and B on the circle, about 70 degrees apart as seen from the centre, and tie a thread between them.
Place pins at three points P1, P2 and P3 on the major arc far from A and B. Tie two threads from each of them, to A and to B.
Tie threads from O to A and O to B to show the angle AOB, and measure it with the protractor.
Measure the angle APB at each of the three points P1, P2 and P3 and write them in a table.
Divide the angle AOB by each of the angles APB and write the ratio, which should always be close to 2.
Move B so that AB becomes a diameter and measure the angle at the edge again. It should be 90 degrees.
Show the three pin positions to a visitor and ask them to guess the angle before measuring.
What you should see
The angle at the centre is about 70 degrees and every point on the major arc shows 35 degrees, with a ratio of 2. For a diameter, the angle at the edge is 90 degrees.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What does the theorem say?
The angle at the centre is twice the angle at the circumference on the same arc.
2What are angles in the same segment?
Angles on the same side of a chord, which are all equal.
3What is the angle in a semicircle?
90 degrees.
4Why do the angles at P1, P2 and P3 match?
Each is half of the same central angle.
Take it further
Show that opposite angles of a cyclic quadrilateral add to 180 degrees with four pins.
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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