E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Measuring and Drawing Angles up to 360°

E2.2About 45 minutes

Learning goals

I can name reflex angles and measure angles up to 360° with a protractor.
I can draw an angle up to 360°.
I can explain why the clockwise angle and the counterclockwise angle between two arms add to 360°.

Words to know

reflex angle
An angle greater than 180° and less than 360°.

full turn
A complete turn of 360°.

protractor
A tool for measuring and drawing angles. Most school protractors are half circles that measure up to 180°; some are full circles that measure up to 360°.

vertex
The point where the two arms of an angle meet.

clockwise / counterclockwise
The way a clock's hands turn, and the opposite way.

Remember from Grade 5

In Grade 5 you measured and drew angles up to 180° with a protractor: put the centre mark on the vertex, line up the baseline with one arm, and read the scale that starts at 0 on that arm. You named angles acute (less than 90°), right (90°), obtuse (between 90° and 180°) and straight (180°).

Let's start with a spin

A figure skater spins around once on the ice. That's a full turn: 360°. If she stops after three quarters of the spin, she has turned 270°.

Angles bigger than a straight angle are everywhere: in turns, in the hands of a clock, in a slice of pizza that's been mostly eaten! An angle between 180° and 360° is called a reflex angle.

Five kinds of angles, each starting from an arm pointing right: acute (45°), right (90°), obtuse (130°), straight (180°) and reflex (240°). The arc shows which side of the arms is being measured.

Two angles between the same arms

Two arms that meet at a vertex make two angles: a small one and a big one. Together they make a full turn, so they add to 360°.

Think of turning from arm A to arm B. You can turn counterclockwise (the small way here) or clockwise (the big way). The two turns add to 360°. The arc in a drawing tells you which angle you're measuring.

Two arms, A and B, meet at a vertex. Turning counterclockwise from A to B is 120° (the small arc). Turning clockwise from A to B is 240° (the big arc). 120° + 240° = 360°.

Measuring a reflex angle with a 180° protractor

Method 1: subtract from 360°.

Measure the small angle between the arms, the normal way.
Subtract from 360°. That's the reflex angle.

Method 2: add to 180°.

Extend one arm backwards with a ruler to make a straight line (180°).
Measure the extra part past the straight line, then add it to 180°.

A protractor measures the small angle between two arms: 140°. The reflex angle on the other side (shown by the arc with the arrow going clockwise underneath) is 360° − 140° = 220°.

Worked example

Example 1: Measure a reflex angle

The small angle between two arms measures 140°. What is the reflex angle?

The small angle and the reflex angle make a full turn: 360°.
360° − 140° = 220°.
Check with Method 2: 220° = 180° + 40°. The arm goes 40° past the straight line, and 180° − 140° = 40° ✓.

220°

Worked example

Example 2: Draw a 300° angle

Draw an angle of 300° with a 180° protractor.

300° is a reflex angle. The small angle that goes with it is 360° − 300° = 60°.
Draw a straight arm. Put the protractor's centre on one end (the vertex) and line up the baseline.
Mark 60° and draw the second arm.
Draw an arc on the outside, the big way around, and label it 300°.

A 60° angle with the arc drawn on the reflex side, labelled 300°.

Worked example

Example 3: Clockwise and counterclockwise

The minute hand of a clock points to 12 and the hour hand points to 3. What is the angle from the minute hand to the hour hand, measured clockwise? Measured counterclockwise?

The 12 numbers split the full turn into 12 equal parts: 360° ÷ 12 = 30° each.
Clockwise from 12 to 3 is 3 parts: 3 × 30° = 90°.
Counterclockwise from 12 to 3 is the rest of the turn: 360° − 90° = 270°.

90° clockwise and 270° counterclockwise. They add to 360°.

Just for fun

Skateboarders and snowboarders call a full spin a "360" because it's a 360° turn. So a "540" is one and a half spins: 360° + 180°!

Mark the arc

Always draw an arc to show which angle you mean. Without the arc, nobody knows if you mean the small angle or the reflex angle. Mathematicians communicate clearly so others can follow their thinking.

Hands-on activity

Angle turns at home

You need: a paper plate (or a circle traced from a pot lid), 2 strips of paper or 2 straws, a brass fastener or a pushpin (ask a grown-up), a protractor, a ruler and a pencil.

Join the two strips at one end through the centre of the plate to make two arms that can turn. (A grown-up should push the fastener through.)
Point both arms at the top of the plate. Turn one arm clockwise a little at a time. Name each angle as it grows: acute, right, obtuse, straight, reflex, full turn.
Stop at a reflex angle. Measure the small angle with your protractor and subtract from 360° to find the reflex angle.
On paper, draw angles of 200°, 270° and 315°. Swap with your grown-up and measure each other's angles.
Stand up and turn your body 90°, 180°, 270° and 360°. Which turns leave you facing the same way you started?

Math talk: "Why do the clockwise and counterclockwise angles always add to 360°?"

Your turn

What kind of angle is 200°?

The small angle between two arms is 75°. What is the reflex angle?

The counterclockwise angle from arm A to arm B is 130°. What is the clockwise angle from A to B?

How would you draw a 310° angle with a 180° protractor?

At 9:00, what is the angle from the minute hand (at 12) to the hour hand (at 9), measured clockwise? Measured counterclockwise?

An angle is a straight angle plus 35° more. What is it, and what kind of angle is it?

More practice: warm-up

How many degrees are in a full turn?

What is a reflex angle?

What kind of angle is 90°?

360° − 100° = ?

Is 180° a reflex angle?

More practice: on your own

At 4:00, the clock hands make a small angle and a reflex angle. Find both.

A wheelchair basketball player spins 270° counterclockwise. What clockwise turn would leave her facing the same way?

The small angle between two arms is 168°. What is the reflex angle?

A reflex angle is 235°. What is the small angle between the same two arms?

Explain why the clockwise angle and the counterclockwise angle from one arm to another always add to 360°.

Challenge

At 8:00, find the angle from the minute hand to the hour hand measured clockwise, and measured counterclockwise.

Three angles fit around a point with no gaps or overlaps. Two of them are 100° and 145°. Find the third angle.

A snowboarder does a spin of 1¾ turns. How many degrees is that? Which way is she facing at the end, compared with the start?

Check your understanding

Quiz

Which angle is a reflex angle?

The small angle between two arms is 110°. What is the reflex angle?

The clockwise angle from arm A to arm B is 300°. What is the counterclockwise angle from A to B?

To draw a 330° angle with a 180° protractor, which angle do you draw first?

How many degrees is a three-quarter turn?

At 2:00, what is the small angle between the clock hands?

For parents and teachers

Big idea: The Ontario curriculum (E2.2) asks Grade 6 students to use a protractor to measure and construct angles up to 360°, and to state the relationship between angles measured clockwise and counterclockwise (they add to 360°). In Grade 5, angles stopped at 180°.

Common difficulties:

Not knowing which angle is meant. Look for the arc. Encourage your child to always draw one.
Reading the wrong protractor scale. This is still common; remind them to start from 0 on the arm they lined up.
Forgetting to subtract. With a 180° protractor, a reflex angle needs one more step: 360° minus the small angle.

How to help: Use clock faces to talk about angles (each hour mark is 30° apart). Turn your bodies 90°, 180°, 270° and 360° together. If your child's school uses a 360° protractor, practise reading it too.