By definition all spheres are identical in shape because a sphere is perfectly round. That's what "sphere" means. In the same way, all perfect squares are identical and all equilateral triangles are identical. Spheres might be larger or smaller, but the shape of all is the same because there is only one kind of sphere.
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Cones, pyramids, and spheres have.
A diameter is a cord in a circle containing the center of the circle. But some circles are sections of spheres. Not all diameters are diameters of spheres.
-- All regular (equilateral) triangles are similar. -- All squares are similar. -- All pentagons are similar. -- All hexagons are similar. . . . etc. Any regular polygon is similar to all other regular polygons with the same number of sides.