Measuring Angles
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- Measuring Angles - Video - HOW TO USE A PROTRACTOR TO MEASURE ANGLES! Watch
Theory
The size of an angle is the amount of turn from one arm to the other. It is measured in degrees (\(^\circ\)) with a protractor. This Year 7 maths page shows how to choose the right scale, read between the numbers, measure reflex angles and find the angles between clock hands.
Benchmarks: a right angle (a square corner) is \(90^\circ\). A straight angle is \(180^\circ\). A revolution (one full turn) is \(360^\circ\), or \(4\) right angles.
A protractor has a centre mark, a base line (the zero line) and two scales. The inner scale starts at \(0\) on one side; the outer scale starts at \(0\) on the other side.
Between the numbers? Each small mark is \(1^\circ\). Start at the last number the arm passed, then count on: \(30\) and \(8\) small marks is \(38^\circ\).
Reflex angles: a protractor stops at \(180^\circ\). Measure the angle between the arms, then reflex angle \(= 360^\circ -\) that angle.
Clocks: \(12\) equal gaps make \(360^\circ\), so each gap is \(30^\circ\). The minute hand turns \(360 \div 60 = 6^\circ\) each minute.
To measure an angle:
- Put the centre of the protractor on the vertex.
- Line up the zero line with one arm.
- Choose the scale that starts at \(0\) on that arm.
- Read where the other arm crosses that scale, and check with an estimate.
How to measure with a protractor
- Estimate: acute, right, obtuse or reflex?
- Place the centre on the vertex and the zero line along one arm.
- Read the scale that starts at \(0\) on that arm.
- Check the reading against the estimate.
Arm \(ED\) is on the left zero, so use the blue scale that starts at \(0\) on the left. Arm \(EF\) crosses \(3\) small marks past \(40\).
| \(40 + 3\) | \(=\) | \(43\) |
\(\angle DEF = 43^\circ\). The angle is acute, so \(137^\circ\) is the wrong scale.
(i) \(\angle MON\)
(ii) \(\angle POM\)
Arm \(OM\) is on the right zero, so read the orange scale.
(i) \(\angle MON = 45^\circ\)
(ii) \(\angle POM = 111^\circ\)
Read both arms on the same (blue) scale, then subtract. Check on the orange scale: \(160 - 55 = 105\).
| \(125 - 20\) | \(=\) | \(105\) |
\(\angle GOH = 105^\circ\)
| \(360 - 118\) | \(=\) | \(242\) |
Reflex \(\angle KLM = 242^\circ\). Check: it is between \(180^\circ\) and \(360^\circ\).
Each gap is \(30^\circ\). At half past, the hour hand is halfway from \(10\) to \(11\). Measure both hands clockwise from \(12\).
| \(10 \times 30 + 15\) | \(=\) | \(315\) |
| \(6 \times 30\) | \(=\) | \(180\) |
| \(315 - 180\) | \(=\) | \(135\) |
| \(360 - 135\) | \(=\) | \(225\) |
The smaller angle is \(135^\circ\), so the reflex angle is \(225^\circ\).
(ii) What is the smaller angle between the hands at \(7\) o’clock?
| \(30 \times 6\) | \(=\) | \(180\) |
| \(5 \times 30\) | \(=\) | \(150\) |
(i) \(30\) minutes at \(6^\circ\) a minute: \(180^\circ\)
(ii) From \(7\) to \(12\) is \(5\) gaps: \(150^\circ\)
Common pitfalls
Frequently asked questions
How many degrees are in a right angle?
\(90^\circ\). A straight angle is \(180^\circ\) and a revolution is \(360^\circ\).
Which scale do you use on a protractor?
The scale that starts at \(0\) on the arm along the zero line.
How do you measure a reflex angle?
Measure the angle between the arms, then subtract it from \(360^\circ\).
How many degrees are between two numbers on a clock?
\(360 \div 12 = 30^\circ\).
How far does the minute hand turn in one minute?
\(6^\circ\).