Raised Fist0

Basic Angle Between Hands

Start learning this pattern below

Jump into concepts and practice - no test required

or
Recommended
Test this pattern10 questions across easy, medium, and hard to know if this pattern is strong

Introduction

In clock problems, one of the most common questions is to find the angle between the hour and minute hands at a specific time. Understanding this pattern is essential because it forms the base for all other clock-related calculations such as right angles, coinciding hands, and reflex angles.

Pattern: Basic Angle Between Hands

Pattern: Basic Angle Between Hands

The angle between the hour and minute hands at any given time is found using:
Angle = |(30 × Hour) - (11/2 × Minutes)|

This formula works because the hour hand moves 0.5° per minute and the minute hand moves 6° per minute. The difference in their positions gives the required angle.

Step-by-Step Example

Question

Find the angle between the hour and minute hands at 3:15.

Solution

  1. Step 1: Write the formula

    Angle = |(30 × Hour) - (11/2 × Minutes)|.

  2. Step 2: Substitute the given time

    Angle = |(30 × 3) - (11/2 × 15)| = |90 - 82.5|.

  3. Step 3: Simplify the result

    |7.5| = 7.5°.

  4. Final Answer:

    7.5°

  5. Quick Check:

    Hour hand moves 0.5° per minute → 15 min × 0.5° = 7.5° shift → matches result ✅

Quick Variations

1. Finding the angle when time is given in different formats (e.g., 6:20 or 9:45).

2. Finding smaller or larger (reflex) angle.

3. Using same formula to determine if hands are at 90° or 180°.

Trick to Always Use

  • Convert time to hour and minute values carefully.
  • Use the direct formula: Angle = |(30H - 11M/2)|.
  • If required, subtract from 360° to get the reflex angle.

Summary

  • The hour hand moves 0.5° per minute, and the minute hand moves 6° per minute.
  • Formula: Angle = |(30 × Hour) - (11/2 × Minutes)|.
  • If the answer exceeds 180°, subtract from 360° to get the smaller angle.
  • Use the same concept for 90°, 180°, and 45° cases.

Example to remember:
At 3:15, angle = 7.5°

Practice

(1/5)
1. Find the angle between the hour and minute hands at 4:20.
easy
A. 10°
B. 100°
C. 50°
D. 130°

Solution

  1. Step 1: Write the formula

    Angle = |(30 × Hour) - (11/2 × Minutes)|

  2. Step 2: Substitute the given time

    Angle = |(30 × 4) - (11/2 × 20)| = |120 - 110| = 10°.

  3. Step 3: Choose the smaller angle

    The computed angle is 10°, which is already the smaller angle.

  4. Final Answer:

    10° → Option A

  5. Quick Check:

    Hour hand = 30×4 + 0.5×20 = 130°; Minute hand = 6×20 = 120°; difference = 10° ✅
Hint: Use |30H - 11M/2| and take the absolute value to get the angle.
Common Mistakes: Not accounting for the hour hand's movement due to minutes.
2. What is the angle between the hands of a clock at 5:30?
easy
A. 75°
B. 15°
C. 150°
D. 165°

Solution

  1. Step 1: Apply the formula

    Angle = |(30 × Hour) - (11/2 × Minutes)|

  2. Step 2: Substitute values

    Angle = |(30 × 5) - (11/2 × 30)| = |150 - 165| = 15°.

  3. Step 3: Interpret

    15° is the smaller angle; reflex would be 360 - 15 = 345° (not needed).

  4. Final Answer:

    15° → Option B

  5. Quick Check:

    Hour hand = 30×5 + 0.5×30 = 165°; Minute hand = 6×30 = 180°; difference = 15° ✅
Hint: Calculate hour and minute positions separately if unsure, then subtract.
Common Mistakes: Assuming the hour hand stays exactly at the hour mark and ignoring its shift.
3. Find the smaller angle between the clock hands at 7:40.
easy
A. 100°
B. 105°
C. 10°
D. 115°

Solution

  1. Step 1: Use the standard formula

    Angle = |(30 × Hour) - (11/2 × Minutes)|

  2. Step 2: Substitute the time

    Angle = |(30 × 7) - (11/2 × 40)| = |210 - 220| = 10°.

  3. Step 3: Confirm smaller angle

    10° is already the smaller angle between the hands.

  4. Final Answer:

    10° → Option C

  5. Quick Check:

    Hour hand = 30×7 + 0.5×40 = 230°; Minute hand = 6×40 = 240°; difference = 10° ✅
Hint: If minutes > 30, expect the smaller angle often to be small; still compute precisely.
Common Mistakes: Confusing the sign when subtracting (use absolute value).
4. At what angle are the hands of the clock at 9:25?
medium
A. 142.5°
B. 137.5°
C. 130°
D. 132.5°

Solution

  1. Step 1: Set up the formula

    Angle = |(30 × Hour) - (11/2 × Minutes)|

  2. Step 2: Substitute time

    Angle = |(30 × 9) - (11/2 × 25)| = |270 - 137.5| = 132.5°.

  3. Step 3: Confirm smaller angle

    132.5° is the smaller angle (less than 180°).

  4. Final Answer:

    132.5° → Option D

  5. Quick Check:

    Hour hand = 30×9 + 0.5×25 = 282.5°; Minute hand = 6×25 = 150°; difference = 132.5° ✅
Hint: Compute hour and minute absolute positions to avoid arithmetic slips.
Common Mistakes: Forgetting to convert minutes into half-degrees for hour hand movement.
5. Find the smaller angle between the hour and minute hands when the time is 11:10.
medium
A. 85°
B. 95°
C. 100°
D. 110°

Solution

  1. Step 1: Apply the clock-angle formula

    Angle = |(30 × Hour) - (11/2 × Minutes)|

  2. Step 2: Substitute values

    Angle = |(30 × 11) - (11/2 × 10)| = |330 - 55| = 275°.

  3. Step 3: Take the smaller angle

    Smaller angle = 360 - 275 = 85°.

  4. Final Answer:

    85° → Option A

  5. Quick Check:

    Hour hand = 30×11 + 0.5×10 = 335°; Minute hand = 6×10 = 60°; difference = 275° → smaller = 85° ✅
Hint: If computed angle > 180°, subtract from 360° to get the smaller angle.
Common Mistakes: Failing to convert the >180° result into the smaller reflex complement.