Packing Intuition Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Packing Intuition.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
Arranging objects of given shapes to fit as many as possible into a bounded region without any overlapping.
How many oranges can you stack in a box? How to arrange them?
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Optimal packing efficiency depends heavily on shape—circles leave gaps that squares would fill.
Common stuck point: Optimal packing problems are often surprisingly hard to prove—even for simple shapes like circles.
Sense of Study hint: Try arranging coins in a square grid, then shift every other row to make a honeycomb pattern. Count how many fit each way.
Worked Examples
Example 1
easySolution
- 1 Step 1: Each coin occupies a 2\,\text{cm}\times 2\,\text{cm} square cell. Number per row = 10/2 = 5. Total coins = 5 \times 5 = 25.
- 2 Step 2: Area of each coin = \pi(1)^2 = \pi cm^2. Total coin area = 25\pi cm^2.
- 3 Step 3: Tray area = 100 cm^2. Packing efficiency = \dfrac{25\pi}{100} = \dfrac{\pi}{4} \approx 78.5\%.
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
hardRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.