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Math · Spatial sense

Angles: Acute, Right, Obtuse, Straight and the Protractor

Name acute, right, obtuse and straight angles, estimate with 45°, 90° and 180°, read the right protractor scale and find clock-hand angles.

Grades 4–5 · 30 practice questions · Original material
Learning objectiveDescribe an angle as the amount of turn between two arms that meet at a vertex; classify angles as acute, right, obtuse or straight; estimate with the benchmarks 45°, 90° and 180°; read a protractor using the scale that starts at 0; and find angles made by clock hands and turns.

The idea

Open a door a little, then a lot. The hinge stays in place while the door turns. An angle measures that amount of turn. We draw an angle as two arms (rays) that start at the same point, called the vertex. The size of the angle is how far one arm has turned away from the other. It does not depend on how long the arms are drawn.

Angles are measured in degrees (°). A full turn is 360°, a half turn is 180° and a quarter turn is 90°. A 90° angle is a right angle, like the corner of a page, and a small square at the vertex marks it. An acute angle is less than 90°. An obtuse angle is more than 90° but less than 180°. A straight angle is exactly 180°: its arms point in opposite directions and form a straight line.

Benchmark angles help you estimate before you measure. Compare with a right angle first: smaller is acute, bigger is obtuse. Half of a right angle is 45°, and two right angles make a straight angle of 180°. An angle that opens a little more than half of a right angle is about 50°.

A protractor has two scales that run in opposite directions. Put its centre mark on the vertex and line up its 0 line with one arm. Then read the scale that starts at 0 on that arm. The other scale at the same spot shows 180° minus the angle, so use your estimate as a check: an angle that looks acute cannot measure 130°.

A clock face is a full turn split into 12 equal steps, so each step from one number to the next is 360° ÷ 12 = 30°. At 3:00 the hands are 3 steps apart: 3 × 30° = 90°.

Worked examples

Four kinds of angles, how to read a protractor, and why arm length does not change an angleTop row: an acute angle of 50 degrees with its vertex and two arms labelled; a right angle of 90 degrees marked with a small square; an obtuse angle of 130 degrees; and a straight angle of 180 degrees. Bottom left: a protractor with its centre on the vertex and its 0 line along one arm. The other arm crosses 50 on the inner scale and 130 on the outer scale. Because the first arm sits on 0 of the inner scale, the angle is 50 degrees; 130 is 180 minus 50. Bottom right: two 40 degree angles, one with short arms and one with long arms, are the same size.An angle measures the turn between two armsCompare with the 90° benchmark: less is acute, more is obtuse, 180° is straight.50°Acuteless than 90°90°Rightexactly 90°130°Obtusebetween 90° and 180°180°Straightexactly 180°vertexarmarmsquare marks 90°Read the scale that starts at 0 on an arm1800150301206090906012030150018013050centre mark on vertex0 line on this armBlue inner scale: 0 sits on the bottom arm,so the angle is 50°. It looks acute, too.Outer scale shows 130° = 180° − 50°: wrong scale.Longer arms, same angle40°40°short armslong arms
Original schematic. The angle is the amount of turn between the arms, so arm length does not change it. On a protractor, read the scale whose 0 lies on one arm; the other scale gives 180° minus the angle.

Example 1 · Use a paper corner to classify

An angle looks smaller than the corner of a page. What kind of angle is it, and what is a sensible estimate if it opens about two thirds of a right angle?

  1. Hold a paper corner (90°) against the angle with one edge on an arm.
  2. The second arm is inside the corner, so the angle is less than 90°: acute.
  3. Two thirds of 90° is 60°.

Acute, about 60°.

Example 2 · Read the correct protractor scale

In the diagram, the bottom arm lies on the protractor's 0 line. The other arm crosses 50 on the inner scale and 130 on the outer scale. What is the angle?

  1. Find which scale has its 0 on the bottom arm: the inner scale.
  2. Follow that same scale to the other arm: 50.
  3. Check with an estimate: the angle is smaller than a right angle, so 50° makes sense and 130° does not.

50°. The outer reading, 130°, is 180° − 50°.

Example 3 · Longer arms, same angle

Two 40° angles are drawn, one with 2 cm arms and one with 6 cm arms. Which is bigger?

  1. An angle measures the turn between the arms.
  2. Both angles have the same turn: 40°.
  3. Longer arms only make the drawing bigger, not the angle.

Neither. Both are 40°.

Example 4 · Clock-hand angle

What angle do the hands of a clock make at 4:00?

  1. Each step from one number to the next is 360° ÷ 12 = 30°.
  2. At 4:00 the minute hand is at 12 and the hour hand is at 4: that is 4 steps.
  3. 4 × 30° = 120°.

120°, an obtuse angle.

Example 5 · Turns as fractions

The minute hand moves from 12 to 4. What fraction of a turn is that, and how many degrees?

  1. From 12 to 4 is 20 minutes out of 60.
  2. 20/60 = 1/3 of a full turn.
  3. 1/3 of 360° is 120°.

1/3 of a turn, or 120°.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Thinking longer arms make a bigger angle.

Why this fails: An angle measures turn. A small drawing and a large drawing of the same corner have the same angle.

Reading the wrong protractor scale.

Why this fails: Each spot shows two numbers that add to 180°. Read the scale that starts at 0 on one arm, then check with an estimate: acute or obtuse?

Putting the protractor's centre mark somewhere other than the vertex.

Why this fails: The scale only measures the turn between the arms when the centre mark sits exactly on the vertex and the 0 line sits on one arm.

Thinking a right angle must stand upright.

Why this fails: A tilted paper corner is still 90°. Turning the whole angle does not change its size.

Calling a straight angle 'just a line' or 'obtuse'.

Why this fails: Two arms that point in opposite directions make a half turn: 180°. Obtuse angles are less than 180°.

Foundation

Foundation practice

Name the angle. Compare it with a right angle (90°) before you choose.

Name the angle

1. An angle measures 35°. What kind of angle is it?

  1. obtuse
  2. right
  3. acute
  4. straight
Name the angle

2. An angle measures 120°. What kind of angle is it?

  1. acute
  2. obtuse
  3. right
  4. straight
Name the angle

3. What kind of angle is the corner of a sheet of printer paper?

  1. right
  2. acute
  3. obtuse
  4. straight
Name the angle

4. Two arms start at the same vertex and point in exactly opposite directions. What is this angle?

  1. obtuse
  2. right
  3. not an angle; it is just a line
  4. straight
Parts of an angle

5. Which part of an angle is the vertex?

  1. the curved arc drawn inside the angle
  2. the longer of the two arms
  3. the point where the two arms meet
  4. the distance between the ends of the arms
Benchmark

6. Which of these measures is an acute angle?

  1. 95°
  2. 90°
  3. 180°
  4. 89°
True or false

7. True or false: a right angle must have one arm pointing straight up and the other pointing straight across.

  1. True
  2. False

Practice

Practice practice

Use turns, clock faces and benchmarks. Each step between clock numbers is 30°.

Turns

8. How many degrees are in a quarter turn?

Turns

9. How many degrees are in a three-quarter turn?

Clock angle

10. At 3:00, what is the angle between the hour hand and the minute hand?

  1. 60°
  2. 90°
  3. 180°
  4. 30°
Clock angle

11. At 6:00, what angle do the clock hands make, and what kind of angle is it?

Clock angle

12. At 2:00, what is the angle between the hands?

Clock angle

13. At 4:00, what kind of angle do the clock hands make?

  1. right (90°)
  2. acute (40°)
  3. straight (180°)
  4. obtuse (120°)
Estimate

14. An angle opens a little more than half of a right angle. Which is the best estimate?

  1. about 50°
  2. about 100°
  3. about 20°
  4. about 140°
Turns

15. The minute hand moves for 45 minutes. What fraction of a full turn is that, and how many degrees?

Think

Think practice

These questions target common mistakes. Explain why the tempting answer fails.

Arm length

16. Angle A has arms 2 cm long. Angle B has arms 8 cm long. Both fit exactly into the corner of a page. Which angle is larger?

  1. They are the same size: both are right angles.
  2. Angle B, because its arms are longer
  3. Angle A, because its arms come to a sharper point
  4. You cannot compare angles with different arm lengths
Protractor

17. Amir puts the protractor centre on the vertex. One arm passes through 0 on the inner scale. The other arm crosses 60 on the inner scale and 120 on the outer scale. The angle looks acute. What does it measure?

  1. 120°
  2. 60°
  3. 180°
  4. 90°
Protractor

18. Leah measures an angle that is clearly smaller than a right angle, but she writes 145°. What most likely happened?

  1. She read the wrong scale; the angle is about 45°.
  2. The angle is 145° because the protractor is exact.
  3. She read the wrong scale; the angle is about 35°.
  4. She placed the vertex wrong; the angle is about 155°.
True or false

19. True or false: if you redraw an angle with longer arms, the angle gets bigger.

  1. True
  2. False
Explain

20. Mia lines up the 0 line with one arm, but she puts the protractor's centre mark at the far end of that arm instead of at the vertex. Why will her reading be wrong?

Order and name

21. Order these angles from smallest to largest and name each type: 95°, 45°, 180°, 30°, 90°.

Turns

22. You face north and make a quarter turn clockwise. Which way are you facing now?

  1. south
  2. west
  3. east
  4. north-east

Challenge

Challenge practice

Combine angles, split turns and watch the hour hand. Show your reasoning.

Clock angle

23. At 3:30, what is the smaller angle between the hour hand and the minute hand?

  1. 90°
  2. 105°
  3. 60°
  4. 75°
Missing angle

24. Two angles share a vertex and together make a straight angle. One of them is 130°. What is the other?

Benchmark

25. A right angle is split exactly in half. What does each part measure?

  1. 45°
  2. 30°
  3. 90°
  4. 60°
Benchmark

26. How many 45° angles fit together, side by side at one vertex, to make a straight angle?

Clock angle

27. At 10:00, what is the smaller angle between the hands?

  1. 300°
  2. 100°
  3. 60°
  4. 30°
Equal parts

28. A round pizza is cut from the centre into 8 equal slices. What angle is at the point of each slice?

Clock angle

29. Which time shows a right angle between the clock hands?

  1. 6:00
  2. 12:00
  3. 1:00
  4. 9:00
Turns

30. A robot turns 90° clockwise, then 90° clockwise again, then 180° counterclockwise. Which way is it facing compared with where it started?

  1. the same direction as the start
  2. the opposite direction
  3. a quarter turn to the right
  4. a quarter turn to the left

Answer key

  1. 1. acute
    35° is less than 90°, so it is acute. It is not obtuse: obtuse angles are more than 90°.
  2. 2. obtuse
    120° is more than a right angle (90°) but less than a straight angle (180°), so it is obtuse. A 120° angle can look 'wide', but it is not straight until the arms form a line.
  3. 3. right
    A paper corner is the classic right angle, exactly 90°. That is why you can use a paper corner to test other angles.
  4. 4. straight
    The arms form a straight line, so the turn is a half turn: 180°. It is still an angle. It is not obtuse, because obtuse angles are less than 180°.
  5. 5. the point where the two arms meet
    The vertex is the shared starting point of the two arms. The arc is only a marking that shows which opening you mean, and the arm length or the gap between the arm ends does not matter.
  6. 6. 89°
    Acute means less than 90°. 89° is just under 90°, so it is acute. 90° is right, 95° is obtuse and 180° is straight.
  7. 7. False
    A right angle is any 90° turn. If you rotate a paper corner, it is still 90°. Turning the whole angle does not change its size.
  8. 8. 90°
    A full turn is 360°. One quarter of 360° is 360 ÷ 4 = 90°.
  9. 9. 270°
    One quarter turn is 90°, so three quarter turns are 3 × 90° = 270°.
  10. 10. 90°
    The minute hand points to 12 and the hour hand to 3. Each step between numbers is 360° ÷ 12 = 30°, and 3 steps × 30° = 90°: a right angle.
  11. 11. 180°, a straight angle
    From 12 to 6 is 6 steps of 30°, which is 180°. The hands point in opposite directions and form a straight line.
  12. 12. 60°
    From 12 to 2 is 2 steps. Each step is 30°, so the angle is 2 × 30° = 60°, an acute angle.
  13. 13. obtuse (120°)
    From 12 to 4 is 4 steps of 30°, which is 120°. That is more than 90° and less than 180°. The 40° option comes from using 10° per number instead of 30°.
  14. 14. about 50°
    Half of a right angle is 45°, so a little more than that is about 50°. About 20° is less than half a right angle, and about 100° and about 140° are both bigger than a whole right angle.
  15. 15. 3/4 of a turn, which is 270°
    60 minutes is one full turn. 45 out of 60 minutes is 3/4 of a turn, and 3/4 of 360° is 270°.
  16. 16. They are the same size: both are right angles.
    An angle measures turn, not arm length. Both fit a paper corner, so both are 90°. Longer arms only make the angle easier to see.
  17. 17. 60°
    Read the scale that starts at 0 on an arm: here, the inner scale. 120° is the outer-scale reading, 180° − 60°, and it would be an obtuse angle, which does not match how the angle looks.
  18. 18. She read the wrong scale; the angle is about 35°.
    An angle smaller than 90° cannot be 145°. The two scales add to 180°, so the other reading at that spot is 180° − 145° = 35°. 45° comes from subtracting wrongly, and 155° does not match an acute angle either.
  19. 19. False
    The size of an angle is the amount of turn between the arms. Making the arms longer does not change that turn.
  20. 20. The centre mark must sit on the vertex. Otherwise the scale is not measuring the turn between the two arms, and the number where the other arm crosses is not the angle.
    The protractor's degree marks are arranged around its centre mark. Only when that centre is on the vertex does each number show how far the second arm has turned from the first.
  21. 21. 30° (acute), 45° (acute), 90° (right), 95° (obtuse), 180° (straight)
    Compare each with 90°: less is acute, exactly 90° is right, between 90° and 180° is obtuse, and 180° is straight. 95° is only a little more than 90°, but it is still obtuse.
  22. 22. east
    Clockwise from north goes north, east, south, west. A quarter turn (90°) moves one step: east. West would be a quarter turn counterclockwise.
  23. 23. 75°
    The minute hand points at 6. The hour hand has moved halfway from 3 to 4, so it is 3.5 steps from 12, or 3.5 × 30° = 105°. The minute hand is at 6 × 30° = 180°. 180° − 105° = 75°. The 90° answer forgets that the hour hand moves; 105° is the angle from 12 to the hour hand, not the angle between the hands.
  24. 24. 50°
    A straight angle is 180°. 180° − 130° = 50°.
  25. 25. 45°
    Half of 90° is 45°, one of the benchmark angles. 30° and 60° would split it into unequal parts.
  26. 26. 4
    180° ÷ 45° = 4. Two 45° angles make a right angle, and two right angles make a straight angle.
  27. 27. 60°
    The hour hand points at 10, 2 steps before 12. 2 × 30° = 60°. 300° is the larger angle measured the long way around; 100° comes from using 10° per number.
  28. 28. 45°
    The slices share a full turn of 360° around the centre. 360° ÷ 8 = 45°.
  29. 29. 9:00
    At 9:00 the hour hand is 3 steps from 12 going the other way: 3 × 30° = 90°. At 6:00 the angle is 180°, at 12:00 it is 0°, and at 1:00 it is 30°.
  30. 30. the same direction as the start
    The two clockwise turns add to 180°. Turning 180° counterclockwise undoes that half turn, so the robot is back where it started.

Related practice

Elapsed time on clocks and schedulesFractions: equivalent fractions and the number lineMath hub