Measuring angles with a protractor

How big exactly is the angle made by the hands of a clock at $4$ o'clock? Asking "how big" demands a number, and the unit we use for angles is the degree, written with a small circle: $°$.

Concept

A full turn , once around, back to where you started , is divided into $360$ equal parts, and each part is one degree. Why $360$? The ancient Babylonians liked the number because it has so many factors ($1, 2, 3, 4, 5, 6, 8, 9, 10, 12, \dots$), making it easy to split a circle into halves, thirds, quarters, fifths, sixths, and so on. So:

• A full turn = $360°$
• A half turn (straight angle) = $180°$
• A quarter turn (right angle) = $90°$
• An eighth of a turn = $45°$

The tool to measure angles is a protractor , a flat semicircle marked from $0°$ to $180°$. To measure an angle:

1. Place the protractor so its centre point (the small notch or cross at the bottom of the flat edge) sits exactly on the vertex.
2. Line up the $0°$ line with one arm of the angle.
3. Read the number where the other arm crosses the curved edge of the protractor.

Most protractors have two scales, an inner and an outer one. Use the scale that starts at $0$ on the arm you have lined up. A common mistake is to read the wrong scale , always double-check by asking "is this angle bigger or smaller than a right angle?" An angle smaller than $90°$ should give a reading less than $90°$.

To draw an angle of a chosen size, say $65°$:

1. Draw a ray $\overrightarrow{AB}$. This is one arm.
2. Place the protractor centre on $A$ and line the $0°$ with the ray.
3. Mark a small point at the $65°$ mark on the curved edge.
4. Remove the protractor and draw a ray from $A$ through that mark.

Measuring angles is a skill, not a piece of magic. It takes a few tries to get accurate, but once your hand learns the moves, you can read any angle in a few seconds.

Worked examples

Example 1. An angle is measured and the protractor shows the second arm at $130°$ (counting from the right $0$). Is the angle bigger or smaller than a right angle?

• $130° > 90°$, so it is bigger than a right angle.

Example 2. Draw an angle of $40°$.

• Draw ray $\overrightarrow{AB}$.
• Place protractor centre at $A$, $0$ on $\overrightarrow{AB}$.
• Mark $40°$ on the curve.
• Draw ray $\overrightarrow{AC}$ through the mark.
• Done , $\angle BAC = 40°$.

Example 3. What angle does the minute hand of a clock turn through in $15$ minutes?

• In $60$ minutes the minute hand turns a full $360°$.
• In $15$ minutes (a quarter of an hour) it turns $\frac{360°}{4} = 90°$.

Example 4. A clock shows $3{:}00$. Find the angle between the hour and minute hands.

• At $3{:}00$, minute hand on $12$, hour hand on $3$.
• The dial is divided into $12$ equal parts, so each part is $\frac{360°}{12} = 30°$.
• From $12$ to $3$ is $3$ parts $= 3 \times 30° = 90°$.

Try it yourself

1. Use a protractor to measure the angle in the corner of your notebook.
2. Draw angles of $30°$, $75°$, and $120°$.
3. What angle does the minute hand cover in $20$ minutes?
4. What angle do the hands form at $6{:}00$?
5. A pie chart shows that $\frac{1}{6}$ of children prefer mangoes. What angle does that slice take up?
6. The angle of a slice of pizza cut into $8$ equal pieces is what?
7. Draw a straight angle and divide it into three equal angles with a protractor. What is each piece?

Activity

Clock-face hunt. Look at a wall clock at five different times during the day. Each time, estimate the angle between the hour hand and minute hand, then check with a protractor (or sketch the clock on paper and measure). Make a small table of times and angles. Can you find a time when the hands form exactly a right angle? Exactly a straight angle?