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Let us say X minutes past 6 o clock, the hands are perpendicular. X < 60 .

hours hand moves 360 deg. in 12 hours.

=> 30 deg in 1 hour. => 1/2 deg in 1 minute

minutes hand moves 360 deg in 1 hour

=> 6 deg. in 1 minute

the angle moved by hours hand in X minutes = X/2 deg.

the angle moved by minutes hand in X minutes = 6 X deg.

The difference must be 90 deg. or 270 deg.

6 X - X/2 = 90 deg. => 11/2 X = 90 => X = 180/11 = 16.36 min

6 X - X/2 = 270 deg => X = 540/11 = 49.09 min

So at 6 hrs 16.36 min and at 6 hours 49.09 min, the hands are perpendicular.

hours hand moves 360 deg. in 12 hours.

=> 30 deg in 1 hour. => 1/2 deg in 1 minute

minutes hand moves 360 deg in 1 hour

=> 6 deg. in 1 minute

the angle moved by hours hand in X minutes = X/2 deg.

the angle moved by minutes hand in X minutes = 6 X deg.

The difference must be 90 deg. or 270 deg.

6 X - X/2 = 90 deg. => 11/2 X = 90 => X = 180/11 = 16.36 min

6 X - X/2 = 270 deg => X = 540/11 = 49.09 min

So at 6 hrs 16.36 min and at 6 hours 49.09 min, the hands are perpendicular.