Work through 2026 AMC 8 Problem 23 on circles and perimeter with a written solution, hints, and a visual explanation.
GeometryLevel 5 · Final stretch
Lakshmi has 5 round coins of diameter 4 centimeters. She arranges the coins in 2 rows on a table top, as shown below, and wraps an elastic band tightly around them. In centimeters, what will be the length of the band?
Diagram 1
A. 2π + 20
B. 5π/2 + 20
C. 4π + 20
D. 9π/2 + 20
E. 5π + 20
Hints
Split the band into its straight segments and its curved arcs, and handle each part separately.
Each straight segment connects the centers of two touching coins — add up those center-to-center distances.
The curved parts together sweep out exactly one full turn (360°), so they form a single circle's circumference — add that to the straight-segment total.
Read the step-by-step solution
Answer: C · 4π + 20
Split the band into straights and arcs
The elastic band is made of straight segments that run between the coins and curved arcs that hug the outer coins. Find each part separately.
Straight parts total 20 cm
Each straight segment is a common tangent between two touching coins, so its length equals the distance between their centers. Connecting the outer centers gives edges 4, 4, 4, and 8: 4 + 4 + 4 + 8 = 20 cm.
4 + 4 + 4 + 8 = 20
Arcs make one full circle
Traveling once around, the arcs turn through a total of 360°, so together they form exactly one coin's circumference: 2π × 2 = 4π. The band length is 4π + 20 cm.
