The angle between the minute hand and the hour hand of a clock when the time…

Key insight: the hour hand moves as minutes pass. Use formulae for positions.

• Hour hand angle (from 12 o'clock) = 30 × (hour + minutes/60).

• Minute hand angle (from 12 o'clock) = 6 × minutes.

• Angle between hands = absolute difference of the two angles; choose the smaller angle (≤ 180°).

Apply to 8:30:

1. Hour hand angle = 30 × (8 + 30/60) = 30 × 8.5 = 255°.

2. Minute hand angle = 6 × 30 = 180°.

3. Difference = |255° − 180°| = 75°.

Therefore the angle between the minute and hour hands at 8:30 is 75°.

Alternate quick formula: angle = |30×hour − (11/2)×minutes|. For 8:30 this gives |30×8 − 5.5×30| = |240 − 165| = 75°.