The angle in a semicircle: measured, traced with a paper corner and proved

The idea

Every angle subtended at the circumference by a diameter is a right angle, and conversely the corner of a right angle sliding against two fixed pins traces a semicircle on them as diameter, which a deductive proof with isosceles triangles explains.

What you need

• A pair of compasses, a ruler with millimetre markings and a 180 degree protractor
• An A3 sheet taped over a corkboard or thick card
• Two drawing pins
• Several A4 sheets (their corners are right angles)
• A plate or saucer about 15 to 20 cm across to trace round (the A4 sheets supply the right angle, and their 21.0 cm and 29.7 cm edges are longer than the traced circle's diameter)

How to do it

1. Draw a circle of radius 6.0 cm and a diameter AB. Mark six points P around the circle on both sides of AB, join PA and PB, and measure each angle APB.
2. Mark one point inside the circle and one outside it, join each to A and B, and measure those angles too.
3. Push the two drawing pins into the A3 sheet 12.0 cm apart. Slide an A4 sheet so that one edge touches each pin and mark the position of its corner; repeat for ten positions on one side of the pins.
4. Mark the midpoint M between the pins and measure the distance from M to each corner mark.
5. Trace round the plate onto paper. Place the corner of an A4 sheet on the traced circle so that both of its edges cross the circle, mark the two crossings and join them; repeat at a second position. Check that the crossing of the two lines is the centre by measuring to the circle in several directions.
6. Write the proof: join P to the centre O; triangles OAP and OBP are isosceles because OA = OB = OP, so the angles at A and B equal the two parts of angle APB, and the angle sum of triangle APB gives 2 × angle APB = 180 degrees.

What you should see

Every angle APB with P on the circle measures 90 degrees within the protractor's ±1 degree. Points inside the circle give angles larger than 90 degrees and points outside give smaller ones: with A and B at (−6, 0) and (6, 0), the point (0, 3) gives 126.9 degrees and (0, 9) gives 67.4 degrees. The ten corner marks all lie 6.0 cm from M within about 1 mm, tracing a semicircle of diameter 12.0 cm. The two joined lines are diameters of the traced circle and cross at its centre, where the distances to the circle agree within 1 to 2 mm. The learner knows it worked when all six angles measure 89 to 91 degrees, the inside and outside angles fall on the correct sides of 90, and the written proof uses only equal radii, isosceles triangles and the angle sum of a triangle.

What changes

This activity lists no variables to change, measure and keep the same.

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

• The angle at the circumference depends on where P is on the circle.
• Only the isosceles triangle in a semicircle has a right angle.
• Measuring six angles proves the theorem.

Safety card

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

• Mathematics K–10 Syllabus (2022), Stage 5 Circle geometry (Path); the page carries the point on proving angle properties of circles, which is this entry; ACARA v9 has no circle geometry anywhere in Years 7 to 10, the whole Year 10 Space set being AC9M10SP01, AC9M10SP02 and AC9M10SP03 and none of them naming circles, so the AC9M10SP01 pointer is the generic deductive-proof description and an inference rather than a match to a content description; page read 2026-09-23, v9 records re-read 2026-09-23MA5-CIR-P-01
• Australian Curriculum v9AC9M10SP01

Sources

The pages the author read to write this activity.

1. curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/fa6eb2f57b
2. amsi.org.au/teacher_modules/Circle_Geometry.html