They are called equidistant points and form points on a sphere for a solid or a circle on a plane figure.
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A spherical surface, with its center at the given point, and its radius equal to the given distance.
They form the sphere whose center is the given point and whose radius is the given distance.
All points in a plane that are equidistant from a given point form a circle. The center of the circle is the given point, and the radius is the constant distance from the center to any point on the circle. Thus, every point on this circle maintains the same distance from the center point.
That's a figure called a "circle". The given distance is the circle's radius, and the point is its center.
It is a sphere.