Estimating and measuring angles

What is it about?

An angle is formed by two rays, the arms $[SA$ and $[SB$ , which start at the same point $S$ . $S$ is called the vertex. Angles are named with Greek letters ( $\alpha$ , $\beta$ , $\gamma$ , …) or with three points: $\alpha = \angle ASB$ . Here – as usual in many school books – $\angle ASB$ is measured anticlockwise from the arm $[SA$ to the arm $[SB$ . The unit is the degree ( $1^\circ$ is one 360th of a full turn).

Type of angle	Size
zero angle	$\alpha = 0^\circ$
acute angle	$0^\circ < \alpha < 90^\circ$
right angle	$\alpha = 90^\circ$
obtuse angle	$90^\circ < \alpha < 180^\circ$
straight angle	$\alpha = 180^\circ$
reflex angle	$180^\circ < \alpha < 360^\circ$
full angle	$\alpha = 360^\circ$

Measuring with a set square in three steps:

Put the zero mark (middle of the long edge) exactly on the vertex $S$ .
Put the edge exactly along one arm.
Read where the other arm crosses the scale – on the scale that starts at $0^\circ$ on the first arm. The set square has two scales: the outer one counts from the right, the inner one from the left.

Reflex angles are greater than $180^\circ$ and do not fit under the set square. Measure the other angle $\beta$ on the other side and calculate:

\alpha = 360^\circ - \beta

Example: $\beta = 130^\circ$ , so $\alpha = 360^\circ - 130^\circ = 230^\circ$ . With a full-circle protractor you can also read reflex angles directly.

Estimating: Compare with a right angle ( $90^\circ$ , “corner of a sheet of paper”) and a straight angle ( $180^\circ$ ). Half a right angle is $45^\circ$ .

Try it

Drag the arm $[SB$ at its handle. When does the type of angle change? Watch the coloured bar.
Set exactly 90°. How is the right angle marked?
Press “Random angle”: the set square now lies to one side and the size is hidden. Slide the set square with its zero mark onto $S$ and turn it at the round handles until the edge lies on an arm. Read off the angle!
For an obtuse angle, place the set square on the other arm. Which scale do you have to read now?
For an obtuse angle, slowly increase $\alpha$ beyond $180^\circ$ . What do the instructions say? How do you measure now?
Measure the same reflex angle with the set square and with the full circle.
Play the estimation game: how many points can you get in ten tasks?

Tasks

Task 1: Types of angles

Name the type of each angle: $35^\circ$ , $90^\circ$ , $179^\circ$ , $181^\circ$ , $270^\circ$ , $360^\circ$ .

Show solution

$35^\circ$ : acute, $90^\circ$ : right, $179^\circ$ : obtuse, $181^\circ$ : reflex, $270^\circ$ : reflex, $360^\circ$ : full angle.

Task 2: Reading the right scale

Load task Measure the angle $\alpha$ . Leon reads $140^\circ$ . What did he do wrong?

Show solution

$\alpha = 40^\circ$ – an acute angle, so the result must be less than $90^\circ$ . Leon read the wrong scale: the two scales add up to $180^\circ$ ( $40^\circ + 140^\circ = 180^\circ$ ). The right scale is the one that starts at $0^\circ$ on the arm along which the edge lies. Show solution

Task 3: Measuring a reflex angle

Load task Find $\alpha$ with the set square.

Show solution

The other angle is $\beta = 65^\circ$ , so $\alpha = 360^\circ - 65^\circ = 295^\circ$ . Show solution

Task 4: Angles on a clock

Through which angle does the minute hand turn in 10 minutes? And in 45 minutes?

Show solution

In 60 minutes it turns through $360^\circ$ , so in one minute through $6^\circ$ . 10 minutes: $60^\circ$ (acute). 45 minutes: $270^\circ$ (reflex). Show 270° – the hand turns clockwise from 12 o’clock ( $[SB$ ) to 9 o’clock ( $[SA$ ).

Task 5: Estimating

Estimate first, then measure: angle 1, angle 2, angle 3.

Show solution

$112^\circ$ (a little more than a right angle), $38^\circ$ (a little less than half a right angle), $205^\circ$ (a little more than a straight angle).

Ideas for the classroom

Use: grade 5, geometric figures and relative positions: estimating, measuring and drawing angles up to $360^\circ$ , types of angles. Well suited for working together at the whiteboard (placing the set square as a class) and for individual practice with the estimation game.
Handling: The set square can be moved in the middle and turned at the round handles or the corners. It snaps to the vertex and to the arms. With “Show the size of the angle” off, students really have to read the size; the highlight then only shows which scale to use.
Model: Set square and grid match: the edge is 16 cm long, one square is 5 mm. Angles are rounded to whole degrees. “Random angle” hides the size and puts the set square to one side.
Estimation game: Tasks alternate between estimating and drawing an angle; the size and the direction of the first arm are random. Points depend on the difference from the correct value: up to $3^\circ$ three points, up to $10^\circ$ two, up to $20^\circ$ one, otherwise none. After checking, the set square is ready for measuring.
Common stumbling blocks:
- Reading the wrong scale (the result and the correct value add up to $180^\circ$ ) – a plausibility check with the type of angle helps.
- Longer arms mean a larger angle. The arms are rays; their drawn length does not matter.
- The zero mark is placed at the corner of the set square instead of on the vertex.
- For reflex angles, the other angle is given as the result.
Notation: The angle $\angle ASB$ is measured anticlockwise from $[SA$ to $[SB$ , so $\angle ASB + \angle BSA = 360^\circ$ .