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.
Step 1: Understanding the Concept:
The question asks us to identify which set of four shapes can be assembled into a complete circle without rotating any of the individual pieces. This requires us to mentally translate the pieces to see if they fit together perfectly.
Step 2: Detailed Explanation:
A full circle is comprised of 360 degrees. We need to check if the pieces in each set can combine to form a continuous circular shape. Since rotation is not allowed, the pieces must fit in their given orientation.
Set A: The red piece is a sector of a circle. The blue piece is also a sector. The yellow piece is a triangle with a curved base. The green piece is a sector with a concave side. These pieces do not appear to have angles that would sum to 360 degrees, and their shapes are mismatched. For example, the straight edge of the yellow triangle cannot fit with the curved edge of the blue sector.
Set B: The shapes are slightly different from set A, but the same problem exists. The straight edges and curved edges do not align in a way that would form a circle. The yellow and green pieces are not simple sectors.
Set C: Again, we have a mix of shapes. The yellow and green pieces are not true sectors of a circle and will not fit with the red and blue sectors to form a perfect circle.
Set D: In this set, all four pieces (red, blue, yellow, and green) are perfect sectors of a circle. Each piece appears to be a 90-degree sector (a quadrant). If we place them together, their central angles will sum to \(4 \times 90^\circ = 360^\circ\). Their curved edges all have the same radius, and their straight edges will meet at the center. By translating them, we can see they will form a complete circle.
Step 3: Final Answer:
The set of pieces in option (D) can form a full circle without rotation.
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Approach Solution -2
Since rotation is not allowed, the only way four pieces close up into a full circle is if their curved edges share one common radius and their straight edges are true radii that can sit flush against each other in their given orientation, so checking each set for these two conditions settles the question quickly.
Set A: the yellow piece is a triangle with a curved base rather than a true sector, so its straight sides cannot lie flush against a neighbouring sector's radius; this set cannot close into a circle.
Set B: the yellow and green pieces are not simple sectors either, and their curved edges do not share the same radius as the red and blue pieces, so gaps or overlaps appear when they are pushed together.
Set C: the yellow and green shapes again deviate from a true sector outline, so their straight edges meet the red and blue pieces at the wrong angle and leave a gap.
Set D: all four pieces are genuine quarter-circle sectors of the same radius, so their straight edges meet cleanly at a single centre point and their curved edges trace one continuous circle. Since \(4 \times 90^{\circ} = 360^{\circ}\), the angles close up exactly.
Only Set D has pieces that are true, equal-radius sectors whose angles add to a full turn, so it is the set that forms a complete circle without any rotation.
