Types of angles — acute, right, obtuse, straight, reflex identification

Question

Classify the following angles: (a) 47°, (b) 90°, (c) 135°, (d) 180°, (e) 250°, (f) 360°. Also, name the type of angle formed by clock hands at 3 o’clock and at 10 o’clock.

Solution — Step by Step

graph TD
    A["Measure the angle"] --> B["0° to 90°?"]
    B -->|"Yes, but not 90°"| C["ACUTE angle"]
    B -->|"Exactly 90°"| D["RIGHT angle"]
    B -->|No| E["90° to 180°?"]
    E -->|"Yes, but not 180°"| F["OBTUSE angle"]
    E -->|"Exactly 180°"| G["STRAIGHT angle"]
    E -->|No| H["180° to 360°?"]
    H -->|"Yes, but not 360°"| I["REFLEX angle"]
    H -->|"Exactly 360°"| J["COMPLETE / FULL angle"]

Angle	Type	Reason
47°	Acute	Between 0° and 90°
90°	Right	Exactly 90°
135°	Obtuse	Between 90° and 180°
180°	Straight	Exactly 180° (forms a line)
250°	Reflex	Between 180° and 360°
360°	Complete	Full rotation

At 3 o’clock: minute hand at 12, hour hand at 3. That is 3 hours out of 12 on the clock face = $3/12 \times 360° = 90°$ . This is a right angle.

At 10 o’clock: minute hand at 12, hour hand at 10. From 10 to 12 is 2 hours = $2/12 \times 360° = 60°$ . This is an acute angle.

Why This Works

Angles measure the amount of rotation between two rays sharing a common endpoint (the vertex). The classification is based purely on the degree measure — there is no ambiguity once you know the number.

Think of it as a door opening: a slightly open door makes an acute angle. A fully open door perpendicular to the wall makes a right angle. A door pushed past perpendicular makes an obtuse angle. A door pushed flat against the opposite wall makes a straight angle. And a door that swings past 180° (imagine a revolving door going further) makes a reflex angle.

The clock is a powerful tool for visualising angles. Each hour mark represents $30°$ ( $360°/12$ ). Each minute mark represents $6°$ ( $360°/60$ ). For exam problems asking “what angle do clock hands make at time X”, this shortcut is very handy.

Alternative Method

A quick visual check: if you can fit a square corner (like a book corner) inside the angle and there is space left, it is acute. If the book corner fits exactly, it is a right angle. If the angle is wider than the book corner, it is obtuse. This physical check helps when you do not have a protractor handy — useful for geometry construction problems in exams.

Common Mistake

Forgetting reflex angles exist. When asked “what is the angle between clock hands at 7 o’clock?”, there are technically two angles — the smaller one (150°, obtuse) and the reflex one (210°). Unless the problem says “reflex angle”, always give the smaller angle. But if the problem specifically asks for the reflex angle, remember it is $360° - \text{smaller angle}$ .