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: Packing intuition arranges copies of a shape to fill a region as fully as possible without overlaps.

Common stuck point: The procedure for packing intuition is the easy part; the trap is assuming circles fill 100% of a box. Asking "Am I fitting many separate copies into a container and judging how much space they fill?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I fitting many separate copies into a container and judging how much space they fill?

Worked Examples

Example 1

easy

Circular coins of radius

1

cm are packed into a

10\,\text{cm}\times 10\,\text{cm}

square tray in a square grid arrangement. How many coins fit and what is the packing efficiency?

Answer

25

coins; packing efficiency

\approx 78.5\%

First step

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

Full solution

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 $= \dfrac{25\pi}{100} = \dfrac{\pi}{4} \approx 78.5\%$ .

Square packing of equal circles achieves efficiency

\pi/4 \approx 78.5\%

. About

21.5\%

of space is wasted as gaps. Hexagonal (honeycomb) packing achieves

\approx 90.7\%

, which is why beehives use hexagons.

Example 2

medium

Compare the packing efficiency of square packing (

\pi/4

) vs hexagonal close packing (

\pi/(2\sqrt{3})

) of unit circles. Which is more efficient and by how much?

Example 3

easy

In a

6\times 6

tray, unit circles are packed in a square grid. How many circles fit, and what fraction of the tray is empty?

Example 4

medium

In hexagonal packing of unit circles, each circle's center is at distance

2

from six neighbors. Find the area of a hexagonal cell around one circle (the Voronoi cell).

Example 5

medium

Three unit circles are packed mutually tangent in a triangular arrangement. Find the side length of the smallest enclosing equilateral triangle, in terms of the circles' radius

1

Example 6

medium

Two unit circles are tangent externally inside a unit-square's diagonal arrangement. If both touch the same long side and one corner, find an arrangement and its total covered area.

Example 7

medium

1 \times 1

square contains four unit-quadrant circles at its corners (each radius

0.5

). What area is uncovered?

Example 8

medium

Four unit circles fit snugly in a

4 \times 4

square. A fifth equal-radius circle is placed in the center gap. Find the largest radius of the center circle.

Example 9

hard

Show why hexagonal packing density is

\pi/(2\sqrt{3})

. Use a unit cell containing one circle.

Example 10

hard

A box

30 \times 20 \times 10

cm is packed with

5

-cm cubes. How many fit, and what fraction of the box is filled?

Example 11

hard

A circular pie pan has radius

10

cm. Estimate how many circular cookies of radius

1

cm fit inside, using packing density

0.907

Example 12

hard

A right triangle with legs

6

and

8

holds non-overlapping unit circles. Estimate the maximum number using hexagonal density (boundary loss is significant).

Example 13

challenge

Show that in 2D, no packing of equal circles can exceed density

\pi/(2\sqrt{3})

, by considering Voronoi cells.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A box is

30

cm long,

20

cm wide,

15

cm tall. How many

5\,\text{cm}

cubes can fit inside?

Example 2

hard

Oranges of radius

4

cm are packed in a square grid on a flat tray that is

40\,\text{cm}\times 40\,\text{cm}

. A second layer is placed on top in hexagonal arrangement offset by

easy

A single circle of radius 1 sits inside a square of side 2. What fraction of the square does it fill (use

\pi \approx 3.14

Example 11

medium

How many circles of radius 1 fit in a single row across a 10-wide tray, and how much width is unused?

Example 12

medium

Why does hexagonal packing fit more circles than square packing in the same area?

Example 13

medium

In a square grid, 4 circles of radius 1 sit in a

4 \times 4

easy

How many

2\times 2

tiles fit in a

10\times 10

floor with no gaps?

Example 24

medium

In a row of circles touching each other, 10 unit circles span what total length?

Example 30

medium

Why does packing efficiency become better when objects are very small relative to the container (for a fixed shape)?

Example 31

hard

In a strip of height

2 + \sqrt{3}

holding two rows of unit circles in zigzag arrangement, how long must the strip be to hold 8 circles (4 per row, with the second row offset by 1 unit)?

Example 32

hard

Two non-overlapping unit circles fit inside a rectangle of width 2. What is the minimum length of the rectangle to accommodate both?

Example 33

hard

Why can packing density depend on container shape even when objects are identical? Give a brief reason.

Example 34

hard

A spherical jar of volume 1000 cm

^3

is filled with marbles of volume 4 cm

^3

each. Approximate the number of marbles using sphere-packing density

0.74

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Related Concepts

Area Volume Tessellation

Background Knowledge

These ideas may be useful before you work through the harder examples.

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