0
0

Basic Angle Between Hands

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

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

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.

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes