The easiest way, even if it is not the best accurate, is the 'trigonometric gun', based on the principle below :
If we know the distance d between the observer O and the launcher L, we are able to calculate the height h by the formulas h = d*tg(alpha). The less is the angle, the best is the précision, so we have to stand quit far from the launcher. Between 50 and 100m seem to be a good distance. 

But this method has some limits. First the sighting is not very accurate, as the estimate of the apogee. Then the locking of the plumb line with the finger doesn't go in the best way for the accurate. Last but not the least, if we look for the opposite picture, we can see that if the trajectory of the rocket is not in a plan perpendicular to the axe LauncherObserver (case of the rockets 2 and 3) the calculation is completely false. 

To compensate for this issue, we must have two observers and use a little more sophisticated device. Each observer have to read 2 angles, the angle alpha as previously, but also the angle Thêta showned on the picture below. 

The L point is the launcher position. The points O1 and O2 are corresponding to the positions of the 2 observers, distant respectively from L1 and L2 of the launcher. The A point is the apogee of the rocket, while the P point is the projection on the ground of the A point. We calculate first the coordinates of P pointby the formulas : Xp = (L1*tg(Théta1)L2*cos(Gamma)*tg(Theta2gamma)L2*sin(gamma))/(tg(theta2gamma)tg(theta1)) Yp=tg(theta1)*Xp+tg(Theta1)*L1 Using these coordinates, we can calculate the height h either with the angle alpha1, either with the angle alpha2. The idéal is to do both calculation, the result must be the same, apart from the uncertainties of measures... The altitude h is given by the following formulas : h = (Yp/sin(Théta1))*tg(alpha1) or h = (Yp(L2*sin(gamma))*tg(alpha2)/sin(gammathéta2) You'd be well advice to make a spreadsheet in Excel or something else or in a programmable pocket calculator to have a fast result. You can also note down the datas and then make a geometric drawing at 1/1000 scale for example (1cm= 10m). In this way, search first the P coordiantes by drawing the lines LO1 et LO2, then, with the angles Théta1 and théta2, the lines O1P and O2P. On another diagram, draw again O1P and its perpendicular in P, then with the angle alpha1, draw the line O1A. to test, draw O2P and its perpendicular in P, then with the angle alpha2, draw the line O2A. In both cases, the height PA must be the same (apart the uncertainties of measures and geometric construction). 
The first trigonometric gun I made, was built with a piece of broom handle, under which I glued a piece of cardboard. On this cardboard, I drew a 1/4 circle with graduation each 5°. In the center of the circle, I fix an axis around which can rotate a metal rod. At the bottom of this metal rod I put a lead ballast.
The broom handle allows to aim at the rocket, and at the apogee, I must lock the metal rod with the finger. Then I can read the alpha angle.
Voici la réalisation pratique, c'est on ne peut plus rustique. Hervé Brégent à réalisé un instrument nettement plus évolué, dont je me suis inspiré très récemment. 

Bellow, you can see the new version which looks like more to a gun. The sight is more accurate. The angle measure is made with a disk which is graduated each 5°. This disk is lead ballasted. When the rocket reach the apogee, I lock the disk with the trigger. Then I can read the angle in front of a marker on the gun. 




Other measure method are availables, for example :
I hope to be able to write a little more on this subject soon.
