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Year 7 Maths Angles

Measuring Angles

20 practice questions 1 video lesson Theory + worked examples

Measuring Angles is about reading and drawing angles in degrees with a protractor. In NSW Year 7 maths it consolidates primary-school skills before the Stage 4 angle rules, using the benchmarks \(90^\circ\), \(180^\circ\) and \(360^\circ\).

You will put the protractor’s centre on the vertex, line up one arm with zero, choose the right scale, count the small marks and check against an estimate. You will also measure reflex angles, draw an angle of a given size and find the angle between the hands of a clock, such as \(120^\circ\) at 4 o’clock or a trickier time like 2:30. This subtopic is part of Year 7 Angles.

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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.

Measuring angle ABC with a protractor Protractor centred on vertex B with arm BA along the zero line on the right and arm BC through 25 on the inner scale (155 on the outer scale). 180160140120100806040200 020406080100120140160180 A C B
Arm \(BA\) sits on the \(0\) at the right, so use the orange scale that starts there: \(\angle ABC = 25^\circ\). The two scales always add to \(180^\circ\) (\(25 + 155 = 180\)).

To measure an angle:

  1. Put the centre of the protractor on the vertex.
  2. Line up the zero line with one arm.
  3. Choose the scale that starts at \(0\) on that arm.
  4. Read where the other arm crosses that scale, and check with an estimate.
\[ \text{reflex angle} = 360^\circ - \text{angle between the arms} \]
reflex angle =360− angle between the arms
To draw an angle. Draw one arm with a ruler. Put the centre on its end and the zero line along it. Make a dot at the size on the scale that starts at \(0\) on that arm, then join the dot to the vertex.
Neither arm on zero? Read both arms on the same scale, then subtract.

How to measure with a protractor

  1. Estimate: acute, right, obtuse or reflex?
  2. Place the centre on the vertex and the zero line along one arm.
  3. Read the scale that starts at \(0\) on that arm.
  4. Check the reading against the estimate.
Example 1 — Arm on the left zero
What is the size of \(\angle DEF\)?
Measuring angle DEF Protractor centred on E with arm ED along the zero line on the left. Arm EF crosses the outer scale at 43, three small marks past 40. 180160140120100806040200 020406080100120140160180 D F E
Solution

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 \(40\).

\(40 + 3\)\(=\)\(43\)
40+3=43

\(\angle DEF = 43^\circ\). The angle is acute, so \(137^\circ\) is the wrong scale.

Example 2 — Two angles from one arm
Find the size of
(i) \(\angle MON\)
(ii) \(\angle POM\)
Measuring angles MON and POM Protractor centred on O with arm OM along the zero line on the right. Arm ON crosses the inner scale at 45 and arm OP crosses it at 111. 180160140120100806040200 020406080100120140160180 M N P O
Solution

Arm \(OM\) is on the right zero, so read the orange scale.

(i) \(\angle MON = 45^\circ\)
(ii) \(\angle POM = 111^\circ\)

Example 3 — Neither arm on zero
What is the size of \(\angle GOH\)?
Measuring angle GOH Protractor centred on O. Arm OG crosses the outer scale at 20 and arm OH crosses the outer scale at 125. 180160140120100806040200 020406080100120140160180 G H O
Solution

Read both arms on the same (blue) scale, then subtract. Check on the orange scale: \(160 - 55 = 105\).

\(125 - 20\)\(=\)\(105\)
125−20=105

\(\angle GOH = 105^\circ\)

Example 4 — A reflex angle
Lena measures \(\angle KLM\) as \(118^\circ\). What is the size of reflex \(\angle KLM\)?
An angle of 118 degrees and its reflex angle Angle KLM of 118 degrees marked with a solid arc between the arms, and the reflex angle marked with a dashed arc the long way round. 118° reflex L K M
Solution
\(360 - 118\)\(=\)\(242\)
360−118=242

Reflex \(\angle KLM = 242^\circ\). Check: it is between \(180^\circ\) and \(360^\circ\).

Example 5 — Clock hands
What is the size of the reflex angle between the hands of a clock at \(10{:}30\) (shown dashed)?
Clock at 10:30 A clock showing 10:30. The hour hand is halfway between 10 and 11 and the minute hand points to 6. The reflex angle between them is marked with a dashed arc. 123456789101112
Solution

Each gap is \(30^\circ\). At half past, the hour hand is halfway from \(10\) to \(11\). Measure both hands clockwise from \(12\).

\(10 \times 30 + 15\)\(=\)\(315\)
\(6 \times 30\)\(=\)\(180\)
\(315 - 180\)\(=\)\(135\)
\(360 - 135\)\(=\)\(225\)
hour hand 315, minute hand 180,315−180=135,360−135=225

The smaller angle is \(135^\circ\), so the reflex angle is \(225^\circ\).

Example 6 — Turning hands
(i) Through how many degrees does the minute hand turn from \(3{:}20\) pm to \(3{:}50\) pm?
(ii) What is the smaller angle between the hands at \(7\) o’clock?
Solution
\(30 \times 6\)\(=\)\(180\)
\(5 \times 30\)\(=\)\(150\)
30×6=180,5×30=150

(i) \(30\) minutes at \(6^\circ\) a minute: \(180^\circ\)
(ii) From \(7\) to \(12\) is \(5\) gaps: \(150^\circ\)

Common pitfalls

Reading the wrong scale. An acute angle cannot read \(155^\circ\). Always check the reading against your estimate.
Centre not on the vertex. If the centre is not on the vertex, or the zero line is not along an arm, every reading is wrong.
Longer arms do not make a bigger angle. The size is the amount of turn, not the length of the arms. Extend short arms with a ruler before you measure.
Measuring a reflex angle directly. A protractor only goes to \(180^\circ\). Measure the other angle and take it from \(360^\circ\).

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?

\(6^\circ\).

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