Practice Packing Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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?

Showing a random 20 of 50 problems.

Example 1

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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 2

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In 3D, what is the densest packing of identical spheres (give density to 3 decimals)?

Example 3

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A square tray 20ร—2020 \times 20 cm is packed with circles of radius 11 cm hexagonally. Roughly how many circles fit using density 0.9070.907? (Each circle occupies area ฯ€โ‰ˆ3.14\pi \approx 3.14.)

Example 4

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

Example 5

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A rectangular shelf is 40ร—2040 \times 20 cm. How many 8ร—58 \times 5 cm books fit if laid flat in a single layer (without rotation)?

Example 6

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Four unit circles fit snugly in a 4ร—44 \times 4 square. A fifth equal-radius circle is placed in the center gap. Find the largest radius of the center circle.

Example 7

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Explain the connection between optimal packing and the isoperimetric efficiency of shapes like the hexagon.

Example 8

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Packing density is the fraction of space filled. If circles fill 78.5% in a square grid, what fraction is empty?

Example 9

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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 11.

Example 10

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Which packs circles more tightly: a square grid or a hexagonal (honeycomb) arrangement?

Example 11

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In a strip of height 2+32 + \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 12

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Compare the packing efficiency of square packing (ฯ€/4\pi/4) vs hexagonal close packing (ฯ€/(23)\pi/(2\sqrt{3})) of unit circles. Which is more efficient and by how much?

Example 13

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

Example 14

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How does packing efficiency change as objects get smaller relative to the container, for a fixed object shape?

Example 15

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A spherical jar of volume 1000 cm3^3 is filled with marbles of volume 4 cm3^3 each. Approximate the number of marbles using sphere-packing density 0.740.74.

Example 16

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Show that in 2D, no packing of equal circles can exceed density ฯ€/(23)\pi/(2\sqrt{3}), by considering Voronoi cells.

Example 17

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A box is 3030 cm long, 2020 cm wide, 1515 cm tall. How many 5โ€‰cm5\,\text{cm} cubes can fit inside?

Example 18

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Why can no packing of equal circles exceed about 90.7% density, no matter how clever?

Example 19

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Why might a manufacturer prefer hexagonal nuts over circular ones for packing/shipping?

Example 20

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A box 30ร—20ร—1030 \times 20 \times 10 cm is packed with 55-cm cubes. How many fit, and what fraction of the box is filled?
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Related Concepts

AreaVolumeTessellation