Practice Rigid vs Flexible Shapes in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A rigid shape cannot be deformed without breaking — its sides and angles are locked. A triangle is always rigid because its three side lengths uniquely determine its angles. A rectangle, by contrast, is flexible: it can collapse into a parallelogram because four side lengths do not fix the angles.
A triangle made of sticks is rigid. A rectangle made of sticks can collapse into a parallelogram.
Showing a random 20 of 50 problems.
Example 1
hardSuppose you have a hinged 3D cube (12 rods, 8 hinged corners). Is it rigid? If not, how many face diagonals are needed?
Example 2
challengeFor a 3D framework with vertices, what is the minimum number of edges needed for generic rigidity (assuming no special configurations)?
Example 3
mediumA square frame is braced with one diagonal. How many rigid triangles does this create?
Example 4
hardA hinged equilateral triangle has each side made of two collinear rods joined at a midpoint hinge (so rods, joints total). Is the figure rigid?
Example 5
mediumA pentagon frame needs braces to become rigid. How many diagonal braces (from one vertex) are needed?
Example 6
mediumIn a triangulated framework, every face is a triangle. Why is this called minimally rigid?
Example 7
mediumA flexible square (side 4) is pushed into a parallelogram. Do its side lengths change?
Example 8
mediumHow many diagonal braces are needed to make a hinged hexagon rigid?
Example 9
easyWhich is rigid: a triangle made of three rods joined at the ends, or a square made of four rods?
Example 10
hardA scissor jack uses two rhombuses linked at a corner. Is this mechanism rigid? Why or why not?
Example 11
easyAre all triangles rigid regardless of their size?
Example 12
challengeA polygon framework with vertices needs how many braces (diagonals) in total to be fully rigid (minimum), and why from a single vertex?
Example 13
hardA rectangular gate is m wide and m tall. It sags toward the latch side. Where should the diagonal brace go: from top-latch to bottom-hinge, or from top-hinge to bottom-latch?
Example 14
mediumA hexagon frame is fully triangulated. Using diagonals from one vertex, how many braces, and how many triangles result?
Example 15
easyA pentagon frame (5 hinged sticks) — is it rigid or flexible?
Example 16
hardA planar framework with vertices is generically rigid if and only if it has at least how many edges (Laman's theorem)?
Example 17
challengeWhy is there an SSS congruence rule for triangles but no equivalent 'side-side-side-side' rule for quadrilaterals?
Example 18
mediumA geodesic dome is built from triangular panels. Why is this naturally a strong structure?
Example 19
easyWhy are bridges and towers often built with triangular frameworks?
Example 20
hardA regular octagon has hinged joints. How many degrees of freedom, and how many braces to make it rigid?