Lines and Angles · Lesson 5 of 9
Degrees and a Homemade Protractor
“Build an angle scale from equal turns and discover what protractor readings mean.”
• Explain one degree as one of 360 equal parts of a full turn. • Derive the measures of half and quarter turns. • Find the angles formed by equal divisions of a circle. • Build a paper angle scale by repeated folding. • Identify an angle bisector and explain why its two parts are equal.
A Unit for Measuring a Turn
Comparing tells us which angle is larger, but it does not tell us how large either one is. To express size with a number, we need a common unit of turn. Imagine dividing a full turn around a circle’s centre into 360 equal small turns. Each part is one degree, written 1°. The degree symbol tells us that the number measures a turn rather than a length.
A unit of angle measure. One degree, written 1°, is one of 360 equal parts of a full turn.
An angle containing 30 of these unit turns measures 30°. You can imagine laying identical tiny openings next to each other around the vertex. The centre stays fixed while the final direction moves farther from the starting direction as more units are included. This is the meaning behind the graduations on a protractor.
Deriving the Reference Measures
A straight angle is half of a full turn, so it contains half of the 360 equal units. A right angle is half of a straight angle, or a quarter of the full turn. We can therefore find their measures by dividing, rather than treating the numbers as facts to memorise without a reason.
Problem
What are the measures of a half turn and half of that half turn?
- 1.Begin with the complete turn: 360°.
- 2.Divide it into two equal parts: 360° ÷ 2 = 180°. Each part is a straight angle.
- 3.Divide one 180° part into two equal parts: 180° ÷ 2 = 90°. Each part is a right angle.
Equal Divisions of a Circle
A sector is a piece of a circle bounded by two rays from the centre and the curved edge between them, like a slice of a round cake. If a circle is split into equal sectors from its centre, their angles together make a full turn. To find one sector angle, share 360° equally among the sectors. The straight boundaries from the centre are the arms; the centre is the shared vertex. Counting equal pieces of the turn is different from measuring distances along the circle’s edge.
Problem
A circle is divided into six equal sectors. Find the angle of one sector and of two neighbouring sectors together.
- 1.All six sector angles fill a full turn, which measures 360°.
- 2.One sector measures 360° ÷ 6 = 60°.
- 3.Two neighbouring sectors make a combined turn of 60° + 60° = 120°. Their shared internal ray does not add an extra turn.
| Equal sectors | Angle of one sector |
|---|---|
| 1 | 360° |
| 2 | 180° |
| 3 | 120° |
| 4 | 90° |
| 5 | 72° |
| 6 | 60° |
| 8 | 45° |
| 9 | 40° |
| 10 | 36° |
| 12 | 30° |
The exact historical reason is not fully known. The Rigveda describes a wheel with 360 spokes, and ancient calendar traditions in India, Persia, Babylonia, and Egypt used 360-day schemes. Babylonian mathematics often counted in groups of 60. These examples show the long history of 360; they do not establish one certain origin for today’s angle unit. A practical advantage is that 360 is the smallest positive whole number divisible by every number from 1 to 10 except 7. It also divides evenly by 12 and 24, so many circle divisions give whole-number degree measures.
Make Your Own Angle Scale
A paper protractor lets you create some degree graduations yourself. Draw and cut out a circle, then fold it into equal halves. The straight fold through the centre is a diameter. Cut along it to keep one semicircle, or half circle. Its straight edge represents a 180° turn, with 0° at the right-hand end and 180° at the left-hand end. The midpoint of that straight edge is the centre for all the turns.
- Fold the semicircle in half so that the two ends of its straight edge overlap. Unfold: the central crease points to 90°.
- Fold again to halve each quarter-turn opening. Unfold: the additional directions are 45° and 135°.
- Repeat one more halving. Unfold: the straight turn is now divided into eight equal 22.5° openings.
- Label the directions from the right-hand baseline as 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5°, and 180°.
Problem
After successive halving folds, the semicircle contains eight equal openings. Find the measure of each and the direction after three intervals.
- 1.The semicircle contains a straight turn of 180°.
- 2.Eight equal openings share it, so each measures 180° ÷ 8 = 22.5°.
- 3.Three intervals from the 0° baseline give 22.5° + 22.5° + 22.5° = 67.5°.
- 4.The same result follows by successive halving: 180° → 90° → 45° → 22.5°.
What an Angle Bisector Does
Each fold above made two equal openings out of one. This is called bisecting an angle: “bi” refers to two, and the division must be into equal parts. A bisecting ray starts at the vertex and passes between the original arms. In a paper-fold model, the crease line contains this ray.
A ray from the vertex that divides an angle into two equal angles. A fold through the vertex can model its direction.
The 45° ray bisects the 90° opening between the 0° and 90° directions. The 90° ray bisects the whole semicircle opening from 0° to 180°. The 67.5° ray bisects the opening between 45° and 90°. For any example, compare the sizes on the two sides; simply pointing somewhere into an angle does not make a ray a bisector.
Problem
An angle measures 90°. A ray divides it into parts measuring 30° and 60°. Is that ray a bisector? What would the parts measure for a true bisector?
- 1.A bisector must make two equal parts. Here 30° and 60° are unequal.
- 2.The two parts do add to 90°, so the ray divides the angle, but does not bisect it.
- 3.For equal parts, calculate 90° ÷ 2 = 45°. Each part would measure 45°.
This folded scale gives only selected directions, separated by 22.5°. It can measure matching openings exactly and help estimate others, but it does not automatically supply a precise reading for every angle. A standard protractor has much finer one-degree graduations.
Quiz
One degree is:
Why is a straight angle 180°?
A circle has five equal sectors. Each central angle measures:
A 180° semicircle is divided into eight equal openings. Each measures:
A ray splits an 80° angle into two equal parts. Each part measures:
Which ray bisects the opening from 45° to 90° on the folded scale?
Practice Problems
- Explain what the degree symbol tells you about a measurement.
- Derive the degree measures of a half turn and a quarter turn from a full turn.
- Find the central angle in circles divided into 3, 8, 9, and 10 equal sectors.
- Build and label the folded semicircle scale. Explain the reason for each new reading.
- List three examples of bisecting rays in your folded protractor, stating the original opening and both equal parts.
- Bisect openings of 60°, 120°, and 150°. Calculate the measure of each half.
- On your scale, find the opening between the 22.5° and 112.5° directions.
- Explain why the folded protractor does not give an exact reading for an angle of 47°.
- A circle is divided into 12 equal sectors. How many sectors make a right angle and a straight angle?
Key Takeaways
• One degree is one of 360 equal parts of a full turn. • Full, half, and quarter turns measure 360°, 180°, and 90°. • For n equal sectors of a circle, one central angle is 360° ÷ n. • Repeated halving builds the 90°, 45°, and 22.5° directions of a paper scale. • An angle bisector makes two equal parts, and a coarse scale has measurement limits.