Measure the angle

$We \space can \space use\space the\space formula\space **30h \space - \space \frac{11}{2}\space m** \space for \space finding \space the \space angle \space between \space minute \space hand \space and \space hour \space hand$ $Here \space ,\space h \space =\space 8 \space and \space m\space=\space 25$ $\Rightarrow \space 30h \space -\space \frac{11}{2}\space m \space =\space 30\times8 \space -\space \frac{11}{2}\times25\space =\space 240\space -\space 137.5\space =\space =102.5^\circ$

$\therefore \space \boxed{Angle\space =\space 102.5^\circ}$

The minute hand completes $1$ revolution in an analog clock in $60$ minutes. One revolution is equivalent to $360°$ . So the rate of the minute hand is $\frac{360°}{60min} = 6°$ $per$ $minute$ . The hour hand completes $1$ revolution in an analog clock in $12$ hours. And $12$ hours is equivalent to $720$ minutes. So the rate of the hour hand is $\frac{360°}{720min} = 0.5°$ $per$ $minute$ .

In short the hour hand moves $0.5°$ per minute and the minute hand moves $6°$ per minute.

At $8:25$ the hour hand has move $8$ hours and $25$ minutes taking $0$ as the origin.

At $8:25$ the minute hand has move $25$ minutes taking $0$ as the origin.

Therefore,

the hour hand has move $(8)(60) + 25 = 505$ minutes or $505 min$ $\frac{0.5°}{min} = 252.5°$

the minute hand has move $25$ minutes or $25 min$ $\frac{6°}{min} = 150°$

smaller angle formed = $252.5 - 150 = 102.5°$

If we don't have a handy formula for the angle between hour hand and minute hand, we can see that from digit 5 on the clock to digit 8 it is an angle of $90^\circ$ .

The hour hand travels $h=\frac{1}{12}\times 360^\circ=30^\circ$ in an hour.

The fraction of the hour is indicated by the minute hand to be $f=\frac{25}{60}=\frac{5}{12}$ .

In that amount of time the hour hand traveled $f\times h=\frac{5}{12}\times 30^\circ=12.5^\circ$ .

At the start of the hour the hour hand was at digit 8, that is $90^\circ$ past digit 5. Now it is $12.5^\circ$ more.So it is at an angle $90^\circ+12.5^\circ=102.5^\circ$ .