Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Protractors: Measuring Angles in Degrees

E2.4About 40 minutes
Learning goals
  • I can explain how a protractor works.
  • I can measure an angle in degrees with a protractor.
  • I can choose the correct scale on a protractor by checking with a benchmark angle.
  • I can classify angles as acute, right, obtuse or straight using their measures.

Words to know

degree (°)
The standard unit for measuring angles. A straight angle is 180 degrees, written 180°.
protractor
A tool for measuring and drawing angles. It is like a ruler for angles.
centre mark
The small mark or hole in the middle of a protractor's straight edge. It goes on the vertex.
baseline
The line on a protractor that goes from 0 on one side, through the centre mark, to 0 on the other side.
scale
A row of numbers on a measuring tool. Most protractors have two scales, one inside the other.

Remember from the last lesson

You measured angles by counting paper wedges. You learned that everyone needs the same unit to compare. Mathematicians around the world agreed on one tiny angle unit called the degree.

Let's start with a ruler for angles

A ruler has many tiny equal spaces, called millimetres. A protractor is a ruler for angles. Its half circle is split into 180 tiny equal angles. Each one is 1 degree, written 1°.

  • A straight angle is a half turn: 180°.
  • A right angle is half of that: 90°.
  • A full turn is two straight angles: 360°.

The 1/8-circle wedge from the last lesson? A full turn is 360°, so each wedge was 360 ÷ 8 = 45°.

18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180centre markbaseline
A protractor. The numbers on the outer scale start at 0 on the left. The numbers on the inner scale start at 0 on the right. The centre mark is in the middle of the straight edge, and the baseline runs along the bottom from 0 to 0.

How a protractor works

The protractor's centre mark sits on the angle's vertex. The lines that fan out from the centre are like the arms of angles of every size, from 0° to 180°.

There are two scales so you can measure an angle that opens to the right or to the left. Each number appears twice, once on each scale. For example, the spot marked 50 on one scale is marked 130 on the other.

Steps to measure an angle

  1. Put the centre mark exactly on the vertex.
  2. Turn the protractor so the baseline lies along one arm.
  3. Find the 0 where that arm crosses the edge. Use the scale that starts at that 0.
  4. Follow that scale around to where the other arm crosses it. Read the number.
  5. Check with a benchmark: is the angle smaller or bigger than a right angle (90°)? Does your number make sense?
18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180
The bottom arm lies along the 0 on the right. Count on the scale that starts at that 0 (the inner scale): the other arm crosses at 50. The angle is 50°.
Worked example

Example 1: An angle that opens to the right

Measure the angle in the picture above.

  1. The centre mark is on the vertex. The bottom arm lies on the baseline and passes through the 0 on the right.
  2. The inner scale starts at that 0. Count up along it: 10, 20, 30, 40, 50.
  3. The other arm crosses the edge at 50 on the inner scale (and 130 on the outer scale).
  4. Check: the angle is smaller than a right angle, so it is acute. 50° is less than 90°, so 50 is right. 130 can't be right!

The angle is 50°.

18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180
One arm lies along the 0 on the left. Use the scale that starts at that 0 (the outer scale). The other arm crosses at 120. The angle is 120°.
Worked example

Example 2: An angle that opens to the left

Measure the angle in the picture above.

  1. One arm runs along the baseline through the 0 on the left.
  2. The outer scale starts at that 0. Count up: 10, 20, 30 … 100, 110, 120.
  3. Check: the angle is wider than a right angle, so it must be more than 90°. 120° makes sense. The other number at that spot is 60, which would be acute, so 60 is wrong.

The angle is 120°. It is obtuse.

Naming angles by their measures

Now we can sort angles exactly:

  • Acute angle: more than 0° and less than 90°.
  • Right angle: exactly 90°.
  • Obtuse angle: more than 90° and less than 180°.
  • Straight angle: exactly 180°.
40°acuteright (90°)90°130°obtuse180°straight
An acute angle of 40°, a right angle of 90° (marked with a small square), an obtuse angle of 130° and a straight angle of 180°.
Worked example

Example 3: The arms are too short

An angle is drawn with arms only 2 cm long. They don't reach the numbers on your protractor. What do you do?

  1. Remember: making the arms longer does not change the angle.
  2. Use a ruler to extend each arm in a straight line, past the edge of where the protractor will sit.
  3. Now measure in the usual way.

Extend the arms with a ruler, then measure. The angle stays the same size.

Check with 90°

A very common mistake is reading the wrong scale. Always look first: is the angle smaller or bigger than a right angle? Then pick the number that makes sense. If you got it wrong, that's OK. Spotting your own mistake and fixing it is what good mathematicians do.

Just for fun

Skateboarders and snowboarders say they did "a 180" when they spin halfway around and "a 360" when they spin all the way around. They are measuring in degrees, just like your protractor!

Hands-on activity

Angle hunt with a protractor

You need: a protractor (they are sold cheaply at dollar stores and school supply stores), a ruler, a pencil and paper.

  1. Set a toy clock or draw a clock face to show 3:00. Measure the angle between the hands. (You should get 90°.)
  2. Measure the corner of a book. Is it exactly 90°?
  3. Open a book or a laptop partway. Hold the protractor's straight edge along one side and read the angle at the hinge.
  4. Look at capital letters on a cereal box or in a newspaper, like A, V, N, K, Z. Trace a few big ones, extend the lines with a ruler, and measure the angles.
  5. For each angle, estimate first, then measure. Record both in a table: Object, Estimate, Measure, Type (acute, right, obtuse or straight).

Math talk: "How did you decide which scale to read?"

18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180
Protractor C, for the questions below.
18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180
Protractor D, for the questions below.

Your turn

1

Look at Protractor C. How many degrees is the angle?

2

Look at Protractor D. How many degrees is the angle?

3

Is the angle on Protractor C acute, right, obtuse or straight? What about Protractor D?

4

How many degrees are in a right angle? A straight angle? A full turn?

5

Kai measures an angle that is clearly smaller than a right angle and writes 140°. What went wrong? What is the real measure?

6

The arms of an angle are too short to reach the protractor's numbers. What should you do?

7

The 1/8-circle wedge from the last lesson: how many degrees is it?

More practice: warm-up

1

A right angle measures how many degrees?

2

A straight angle measures how many degrees?

3

Is a 72° angle acute, right or obtuse?

4

Is a 150° angle acute, right or obtuse?

5

Where does the protractor's centre mark go?

More practice: on your own

1

An angle is exactly 3 of the 1/8-circle wedges. How many degrees is it?

2

An angle is exactly half of a right angle. How many degrees is it?

3

The arm of an angle crosses where the scales say 60 and 120. The angle is clearly wider than a square corner. What is its measure?

4

Put these angles in order from smallest to largest and name each type: 95°, 89°, 179°, 12°.

5

Lucas lined up the baseline with one arm, but the vertex was not on the centre mark. Will his reading be correct? Explain.

Challenge

1

A clock face is a full turn of 360° split into 12 equal spaces. How many degrees is each space? How big is the angle between the hands at 2:00?

2

Two angles sit side by side and together make a straight angle. One of them is 65°. What is the other?

3

Zoe says, "Every angle on a protractor has two numbers, and they always add to 180." Is she right? Test it with an example.

Check your understanding

Quiz

1.

How many degrees is a right angle?

2.

What goes on the protractor's centre mark?

3.

An angle's arm crosses where the scales say 30 and 150. The angle is smaller than a right angle. What is it?

4.

Which angle is obtuse?

5.

Why do most protractors have two scales?

6.

How many degrees are in a full turn?

For parents and teachers

Big idea: The Ontario curriculum (E2.4) asks Grade 5 students to explain how protractors work, use them to measure and construct angles up to 180°, and use benchmark angles to estimate. This lesson focuses on what the protractor is and how to measure. The next lesson covers drawing angles and estimating.

Common difficulties:

  • Reading the wrong scale. This is a very common error. Teach the habit: "Is it bigger or smaller than a right angle?" before reading.
  • Vertex not on the centre mark. Some protractors have the centre mark above the bottom edge. Show your child exactly where it is on their protractor.
  • Arms too short to read. Extending arms with a ruler is allowed and does not change the angle.

How to help: A protractor is inexpensive and worth having at home. Ask your child to teach you how to measure an angle. Explaining the steps out loud strengthens understanding.