You have to know these things when you're king.
Imagine being out for a ride on a beautiful day, minding our own business, when suddenly we find ourselves riding through the middle of an archery range. It could happen to any of us. How dangerous is it for our helmeted heads to be on the receiving end of an arrow? Edwin Sarkissian has performed a public service and demonstrated in this video.
He starts with a traditional recurve bow, quite similar to the antique Hoyt Pro Medalist that I shoot. The first shot is straight toward the visor, which we know protects your head from little more than road debris and bird droppings. Predictably, the arrow penetrates the visor somewhere between three to five inches. Honestly, that's far less than I expected it to penetrate based on my own experience of stabbing things from far away. After rotating the helmet onto its side he shoots at the top of it. While the tip definitely leaves a mark, it bounces right off. It seems that the plastic of this particular helmet is safe from piercing damage of an arrow moving 180 feet per second. Don't worry, though, this is just the beginning.
Edwin breaks out a compound bow, something that we in the traditional archery community refer to as "training wheels." It is certainly more powerful and accurate, however, as shown by two arrows penetrating in one end, out the other, and even slightly into a second helmet sitting next to it. Then the even more powerful crossbows come out, and these helmets don't stand a chance.
Nor should they be expected to. Helmets are designed to protect your head from a major impact against a solid object, such as the ground or a guardrail. In Dungeons and Dragons terminology, they are designed to protect you from bludgeoning damage. They are not designed to protect you from the piercing damage of an arrow. Certainly, a stronger shell, as well as shapes that would tend to deflect the arrow, would help with this, but how often do riders find themselves on the pointy end of a flying arrow? Hopefully never.
Still, if you ever do find yourself riding through the middle of a Society for Creative Anachronism archery tournament, your helmet should keep your brain intact from the arrows they'd be shooting from traditional bows. I still would not recommend doing so, however. Not only would the arrows easily penetrate the rest of your riding gear, but there are also often big burly fighters nearby to take you down.