Which set of pieces can form a full circle, if rotation of pieces is not permitted?
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In shape assembly puzzles, first check the fundamental geometry. To form a circle, all outer boundaries must be circular arcs of the same radius, and all inner vertices must meet at a single central point. All pieces must be true sectors.
A full circle needs pieces whose central angles add up to exactly \(360^{\circ}\), so a fast check is to estimate each piece's angle and add them up for every set.
In sets A, B and C, at least one piece is not a clean sector, it has a straight edge cutting across a curve or an uneven width, so its angle is not well defined and the pieces cannot close up into a smooth circle no matter how they are pushed together.
In set D, all four pieces look like equal quarter-slices, each contributing \(90^{\circ}\), and \(90^{\circ} + 90^{\circ} + 90^{\circ} + 90^{\circ} = 360^{\circ}\), which is exactly a full turn.
Since only set D gives four genuine sector angles that sum to \(360^{\circ}\) with matching radii, it is the set that completes the circle.
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Approach Solution -2
A different check is to compare the curve radius of each piece within a set, since pieces from a single circle must all curve at the same distance from the centre; if even one piece has a tighter or wider curve than the rest, the set cannot form one smooth circle.
Set A: the yellow triangle's curved base has a different curvature to the red and blue sectors, so it would leave either a gap or an overlap along the rim.
Set B: the green piece's concave edge curves the opposite way to the other three, so it cannot sit flush along the same circle.
Set C: the yellow and green pieces again show a curvature mismatch against the red and blue sectors, breaking the smooth rim.
Set D: all four curved edges share the same radius, so laid edge to edge they trace one unbroken circular rim with no gap or overlap.
Only set D keeps a single consistent radius across all four pieces, which is what lets them close into a full circle without being rotated.
