Work through 2026 AMC 8 Problem 3 on perimeter with a written solution, hints, and a visual explanation.
GeometryLevel 1 · Warmup
Haruki has a piece of wire that is 24 centimeters long. He wants to bend it to form each of the following shapes, one at a time:
A regular hexagon with side length 5 cm.
A square with area 36 cm².
A right triangle whose legs are 6 and 8 cm long.
Which of the shapes can Haruki make?
A. Triangle only
B. Hexagon and square only
C. Hexagon and triangle only
D. Square and triangle only
E. Hexagon, square, and triangle
Hints
The wire is 24 cm long — figure out the perimeter each shape actually needs.
Compute the hexagon's perimeter (6 × side), the square's side length from its area, and the triangle's hypotenuse from its two legs.
Compare each shape's perimeter to 24 cm — only the shapes that match exactly can be made.
Read the step-by-step solution
Answer: D · Square and triangle only
Check each shape's perimeter
Hexagon: 6 × 5 = 30 cm (too long). Square with area 36: side = 6, perimeter = 4 × 6 = 24 cm (works). Right triangle with legs 6 and 8: hypotenuse = 10, perimeter = 6 + 8 + 10 = 24 cm (works).
Conclusion
Only the square and triangle can be made with 24 cm of wire.