In hyperbolic geometry, are similar polygons with different areas possible?

• A. Yes
• B. No
• C. Only for certain polygons
• D. Only for triangles

Answer: (C)

In hyperbolic geometry, a polygon’s area is fixed by its angle defect: area equals (n − 2)π minus the sum of its interior angles, where n is the number of sides. If two polygons are similar, they have corresponding angles that are equal, and they have the same number of sides. That means the sum of their interior angles is the same, so the angle-based formula gives the same area for both. Because similarity doesn’t allow a change in the angle structure while preserving the same side count, you cannot get different areas from similar polygons. So similar polygons cannot have different areas in the hyperbolic plane.