Angle Measurement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Angle Measurement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Angle measurement is the process of determining the size of an angle in degrees using a protractor or by calculation.

A protractor is like a ruler for turns — it tells you exactly how much one line has rotated from another.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Measuring an angle means anchoring the vertex and reading the correct scale.

Common stuck point: The procedure for angle measurement is the easy part; the trap is placing the protractor center away from the vertex. Asking "Is the protractor center on the vertex and one ray on 0?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the protractor center on the vertex and one ray on 0?

Worked Examples

Example 1

medium
On a protractor, one ray of an angle is aligned with 0°0° and the other ray crosses at 65°65°. What is the angle measure?

Answer

65°

First step

1
Start counting from the 0°0° aligned ray.

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Example 2

medium
A protractor reading shows one ray at 40°40° and the other at 155°155°. What angle do they form?

Example 3

hard
Find the angle between clock hands at 2:30.

Example 4

hard
Find the angle between clock hands at 9:00.

Example 5

hard
A regular hexagon's interior angle is how many degrees?

Example 6

hard
At what time after 12:00 do the hour and minute hands first overlap?

Example 7

challenge
Convert 2π3\tfrac{2\pi}{3} radians to degrees.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A protractor's center hole must be placed on which part of the angle?

Example 2

easy
Which is bigger: a 45° angle or a 75° angle?

Example 3

easy
Does making the rays of an angle longer change the angle measure?

Example 4

easy
Classify a 135° angle.

Example 5

medium
Two angles together make a straight line. One is 52°52°. Find the other.

Example 6

medium
Two angles meet at a corner and form a right angle. One is 37°37°. Find the other.

Example 7

medium
A pie chart slice represents 13\tfrac{1}{3} of the data. What is its central angle?

Example 8

medium
A turn of −90°-90° (clockwise quarter) is the same as which positive turn?

Example 9

medium
An angle and its reflex partner add to 360°360°. If the angle is 145°145°, what is the reflex?

Example 10

hard
Two supplementary angles are in the ratio 2:32:3. Find the larger angle.

Example 11

hard
Two complementary angles differ by 14°14°. Find the smaller angle.

Example 12

hard
An angle is three times its supplement. Find the angle.

Example 13

challenge
A regular polygon has each interior angle 150°150°. How many sides does it have?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

angles