The biggest difference between concave and convex is in their shape. Concave is used to describe any surface that curves inward. Convex, on the other hand, is used to describe a surface with an outline or surface that curves outward. The terms are used as adjectives to describe lens types, mirrors, polygons, etc.
A concave surface is like the inside of a circle, whereas a convex surface is like the outside of a circle or sphere.

Concave | Convex | |
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What is it | Concave is an adjective that describes a surface that curves inward, or that is thinner in the middle than at the edges. |
Convex describes a surface that curves outward, or is thicker in the middle than the edges. |
Mirror | A concave mirror is a spherical mirror in which the reflecting surface and the center of curvature are on the same side of the mirror. In other words, if the mirrored part is the inner part, the mirror is concave. | A convex mirror is a spherical mirror in which the reflecting surface and the center of curvature are on opposite sides of the mirror. In other words, if the mirrored part is the outer part, it is a convex mirror. |
Lens | Convergent. A concave lens is thinnest in the middle and thickest at the edges. |
Divergent. The convex lens is thickest in the middle and thinnest at the edges. |
Meaning | Concave means "hollow or rounded". | Convex means "curved". |
Polygon | A polygon with one or more interior angles greater than 180 degrees. | All interior angles are less than 180 degrees. |
Examples | The side mirror of cars, the inner surface of most eyeglasses, the lens of binoculars. | The outer surface of an eyeglass lens, camera lens, magnifying glass, and telescope lens. |
What is concave?

Concave is an adjective that describes a surface that curves inward, or that is thinner in the middle than at the edges. Examples: The side mirror of cars, the inner surface of glasses, the lens of binoculars.
What is convex?

Convex describes a surface that curves outward, or that is thicker in the middle than the edges. Examples: The human eye, the outer surface of eyeglass lenses, camera lenses, loupes, and telescope lenses.
Concave lenses and convex lenses
A concave lens is thinnest in the middle and thickest at the edges. On the other hand, a convex lens is thicker in the middle and thinner at the edges. The convex lens is known as a converging lens, while the concave lens is known as a diverging lens.
Concave and convex mirrors
Concave and convex are also two types of spherical mirrors. A concave mirror is a spherical mirror in which the reflecting surface and the center of curvature are on the same side of the mirror. A convex mirror is a spherical mirror in which the reflecting surface and the center of curvature are on opposite sides of the mirror.
Concave polygons and convex polygons
The terms concave and convex are also used in the context of polygons. A polygon with one or more interior angles greater than 180 degrees is called a concave polygon. On the other hand, a polygon where all interior angles are less than 180 degrees is known as a convex polygon.
Summary
Concave surfaces curve inwards, convex surfaces curve outwards.