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In the Harry Potter books, Metamorphmagi are people with the innate ability to change their own shape at will. Tonks is the main example of a Metamorphmagus, and she is shown changing the shape of her nose, the colour and shape of her hair, and once even taking the form of an entirely different old lady.

What are the limits imposed on this ability, though? Do Metamorphmagi have to maintain some human form? Are they free to morph into any anthropomorphic creature, such as a goblin or a house-elf? What about any animal or plant? Can they go even further, and transform into inanimate objects such as an oversized armchair like Slughorn does through Transfiguration? Can they get as big or as small as they want?

In Order of the Phoenix, Tonks explains to Harry that she can change her appearance at will, not her shape.

‘I’m a Metamorphmagus,’ she said, looking back at her reflection and turning her head so that she could see her hair from all directions. ‘It means I can change my appearance at will,’ she added, spotting Harry’s puzzled expression in the mirror behind her.

_{Order of the Phoenix - pages 51-52 - Bloomsbury - chapter 3, The Advance Guard}

If there is an example of Tonks changing into another race, such as a goblin or house-elf, I personally cannot recall it from canon.

I don't think being a Metamorphmagi extends to animals because turning oneself into an animal is specifically called an Animagus.

My interpretation of canon is that a Metamorphmagus can change their human appearance. So, yes, Tonks can change her appearance to look like an old lady or Mundungus Fletcher or Aunt Petunia or Dumbledore. But she cannot change her appearance to look like Griphook, Dobby, Firenze, Peeves, or the Weasley clock.

As well, baby Teddy Lupin, who is a Metamorphmagus, in his baby picture is not shown turning into a goblin or chair -- his hair is shown changing colors. As children in the wizarding world do not have control over their magic (Tom Riddle being the exception), our best bet at seeing whether a Metamorphmagus could turn into a broomstick or a Grindylow would have been with Teddy Lupin. Yet he did not turn into anything non-human.

This is only going by what we know from the books. Perhaps a Metamorphmagus has powers we don't know about, but J.K. Rowling hasn't disclosed any to date.

On the Harry Potter Wiki, it states that:

Unlike Animagi, Metamorphmagi can take a wide variety of forms, changing sex and age, potentially looking like anyone at all, or even just changing a part of their appearances, such as the shape of their noses. Animagi also transform at will, but only into one animal form which is determined by their inner traits. It is never revealed if Metamorphmagi can take the full form of an animal, however it is very likely to be true if practiced hard enough and with the right skills

Also, a Metamorphmagus is born, not made; it is not a skill that can be learned. There also seems to be a degree of heritability, as both Nymphadora Tonks and her son, Teddy Lupin, were Metamorphmagi, although both of Tonks' parents, Andromeda and Ted, were not. The ability presents very early in life, as Teddy Lupin's hair changed colour when he was a newborn infant, as did his mother's.

Not sure how true this is, but most other Google results for 'Metamorphmagus' say very little on the topic. Fact is, we really don't see much of Tonks in the books, and while they may not use her abilities much in the Order, we've never seen how she uses it as an Auror and the only thing that's ever really said about it is in the beginning of the 5th book. It's not a topic covered much in the books so really, its open to speculation. If she could only change her appearance and not her shape, she wouldn't be able to change the shape of her nose. She probably just simplified her description for Harry. If she can do a pig nose, why not a hand into a paw? If she can change her hair colour, style and length, why not grow a beard? Can she change her skin into another texture or substance? Could she be like a chameleon and blend in with brick, or bark, or leather? Can her hair turn into feathers or leaves? We know very little about metamorphmagi in canon. As stated earlier, it really is open to speculation and, most importantly, up to your imagination. I doubt, however, that she could completely turn into an inanimate object. Animal, perhaps (probably mammal? Something with at least a passing similarity to humans, like a cat or a dog. I doubt she could turn into a goldfish or an ant). Plant, maybe appearance and shape (like a tree, with bark skin and leaf hair and make her body look like a tree shape) but I doubt she could turn into a daisy. A vase or a chair? Highly unlikely, in my opinion.

Another interesting thing to consider is an idea I got from some fanfictions. The idea is that there are various degrees of ability, some Metamorphmagi can only change hair colour, others hair and nail length, right up to those who can completely change age (even turn into a child), shape, sex, race etc.

But we'll probably never really know. Has anyone ever asked J.K.Rowling?

I don't have the 5th book with me at the moment, but I do remember Hermione or Ginny saying:

"Do the one that looks like a pig's snout!"

She said 'looks like' implying that the nose look as close to a pig's snout as is possible for a human. I can further argue this with Harry thinking she looks like a female Dudley. We know Dudley cannot look non-human so I assume it looks close to a pig's snout. We already have Animagi, so it's not very likely that Metamorphmagi could change into animals.

I think Metamorphmagi couldn't change into inanimate objects, because they (the objects) lack the ability to think and therefore could not change back if they hypothetically changed into one, so it isn't likely they'd even have that ability because Teddy would have demonstrated it due to his aforementioned lack of ability to control his powers.

Metamorphagi are able to easily change their hair color, shape, and length. If I remember correctly, in book five, Tonks transformed her entire body into that of a wizened old lady's. She was able to change her whole self, including her night and voice. Further more, if she can morph her own nose into a pig's snout, why wouldn't she be able to turn her teeth into tusks or her feet into hooves?

I doubt that she would be able to transform into inanimate objects or animals. It is said it looks like a pig snout, not is one. The movie misrepresents this I believe, making it seem like a pig snout. Yes, if it really was a pig snout, then why not an animal? And, objects do not possess the ability to think, so how would it be possible to change back to a human form? J.K. Rowling has not revealed any specific information, however, so we cannot be 100% sure.