The sharpness of mammal teeth and claws

It recently occurred to me that cats’ and dogs’ claws seem to be sharper than their teeth. I would have expected a carnivore’s teeth to be as sharp as possible, but based on what I have seen, this doesn’t seem to be the case. I now have several questions:

Is it generally true that carnivorous mammals have claws sharper than their teeth?
What’s the sharpest known tooth? What’s the sharpest known claw?
How well does tooth enamel hold a point? It’s supposed to be the toughest material produced by the body, but brittle materials can often hold sharper edges than tougher materials–for example, glass vs. steel.

Cat’s claws are “self-sharpening”. Dog’s claws tend to get worn to a chisel edge, not nearly as sharp as a cat’s. Both animal’s teeth are probably as sharp as possible given the strength of materials constraint.

Not necessarily if the carnivore can’t regrow lost or damaged teeth, which cats and dogs can’t. They can regrow lost or damaged claws, so it’s not such a big deal if their claws are sharper than they are strong and they lose a claw. A wild carnivore, on the other hand, is in obvious trouble if it loses the teeth it needs to kill prey or the teeth it needs to rip chunks out of the prey.

I would look to animals that can regrow lost teeth, such as sharks, for the sharpest teeth.

Is there actually a way to measure sharpness of teeth or claws?

The sharpness of blades is certainly tested however I have not been able to come across an ISO standard but I have no doubt there will be one.

Ah found it ISO 8442.5

I think you need to also consider the force that the cutting edge would need to withstand, along with the service life too.

Does dog hunting behavior involve the claws? I’ve never noticed my dog to do anything with his claws to hurt anything, while my cats do that disembowelling move in play all the time. Unsurprisingly, the cat claws are a whole lot sharper than the dog ones.

Cats use their claws to capture prey, and in that operation sharpness matters a lot (dull claws = lost prey = no meal). Once captured, they hang on with both teeth and claws.

But large prey isn’t really killed with teeth directly: a lion or tiger typically kills by twisting the neck to one side (away from the hooves, so they can’t cause damage). The animal dies of asphyxiation. Once it’s dead, there is plenty of time to rip off flesh without the need for extremely sharp teeth.