In the phrase 'area of a shape', both area and shape are nouns. In your mind, add 'the', it helps to show that they are nouns: 'the area of the shape'.
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The word 'area' is a noun; a word for the amount of space that the surface of a place or shape covers; a particular subject or type of activity; a part of a city, town, or country; a word for a thing.
Yes
The word ring cannot be an adverb. It can be a noun (object, shape) or a verb (to make a ringing sound).
The noun 'growth' is an abstract noun as a word for the process of increasing in amount, value, or importance; a word for a concept.The noun 'growth' is a concrete noun as a word for an amount in a size increase or a thing that has grown; a word for a physical thing.The noun 'ability' is an abstract noun, a word for talent, skill, or proficiency in a particular area; the quality or state of being able; a word for a concept.There are no abstract noun forms for the concrete nouns 'magazine' and 'elbow'.
The noun 'addition' is an abstract noun as a word for the action of adding something to something else; the process of adding two or more numbers or amounts together.The noun 'addition' is a concrete noun as a word for something or someone more that is included; a room or area that is added to a building.