Constructing a 60°, 30° or 15° Angle

Constructing a ${30}^{\circ }$ Angle

If you need to find an angle that is the half the size of another angle, you use the bisection technique. So, if you want to construct a $30$${}^{\circ }$ angle, you just construct a $60$${}^{\circ }$ angle, and then bisect it. You just make a cross midway between the two sides of the angle and draw a line from the vertex through this cross.

To do this, set the point of your draft compass at $A$ and draw an arc to the right of your existing $60$${}^{\circ }$ arc. Keep the distance of the legs of your draft compass, and set the point of the compass at the intersection of the two crossing arcs and the $60$${}^{\circ }$ line. From this point, draw a new arc intersecting the arc you marked when the point was set at $A$. These intersecting arcs should make a cross. Finally, to bisect the $60$${}^{\circ }$ angle, draw a straight line from $P$ until this line intersects the newly created cross.

The two new angles are both $30$${}^{\circ }$, because $30+30=60$.

Constructing a ${15}^{\circ }$ Angle

To make a $15$${}^{\circ }$ angle, you’ll once again need to construct an angle that is half the angle you have just constructed, so you only need to use the bisection technique again. In other words, all you need to do to construct a $15$${}^{\circ }$ angle is bisect a $30$${}^{\circ }$ angle.

That means you just make a cross midway between the two sides of the angle using a drafting compass and draw a line from the vertex through this cross, using the same instructions as before. The two new angles are both $15$${}^{\circ }$, because $15+15=30$.