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From the Cambridge Dictionary

rounded (adjective): round or curved src

round (adjective): shaped like a ball or circle, or curved src

I've seen both "round shape" and "rounded shape" many times.

Ngram Viewer shows both are common.

enter image description here

Do they mean the same thing? Are they interchangeable for any cases? Is the consideration for formal vs. informal involved?

For example, "round stone" means the shape of the stone is round. In this particular situation, what's the difference between "round shape" and "rounded shape"?

I don't see the difference between google image: rounded stone and google image: round stone

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    My two cents: Round shape may refer to objects that are almost spherical but rounded may refer to objects like eggs or pebbles that are irregular ovoids. – Bruce Murray Jun 16 at 10:28
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They mean the same thing in some contexts, especially when describing aspect/shape/form, but not in others.

Dictionary.com gives the following alternative meaning:

fully developed, perfected, or complete; diversified and well-balanced (sometimes used in combination): a well-rounded education; a rounded character.

As a matter of preference I would also make a small distinction between the two words where shape is concerned, as in:

She picked up a round stone from the beach

The sculptor gave the orb a rounded form
(implying some action to produce the shape)

https://www.dictionary.com/browse/rounded

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"round shape" -refers to something which is circular form in nature.

while "rounded shape" -represents to a thing which can be characterize in different forms but was intendedly designed and curved as a spherical object at some point. Like a rounded shape Satellite Dish, that attached in a property to serve as an Antenna.

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