What is the maximum distance between two points in a square of side length 1?

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In summary, the maximum distance between two points in a square of side length 1 is the diagonal of the square, which can be calculated using the Pythagorean theorem. This distance cannot be greater than the diagonal and remains the same regardless of rotation. The maximum distance also equals the diameter of a circle with radius 1, and is longer than the perimeter of the square.
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Ackbach
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Here is this week's POTW:

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Show that if $5$ points are all in, or on, a square of side length $1$, then some pair of them will be no further than $\dfrac{\sqrt{2}}{2}$ apart.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to greg1313 and Fallen Angel for their correct solutions. greg1313's solution follows:

Construct four quarter circles with radius $\dfrac{\sqrt2}{2}$, each with its centre on a vertex of the square. Notice that there is no area on the square that is not contained by, or on, a quarter circle. Hence one cannot place more that four points in or on the square without such a point being at most $\dfrac{\sqrt2}{2}$ from another point.
 

Related to What is the maximum distance between two points in a square of side length 1?

1. What is the maximum distance between two points in a square of side length 1?

The maximum distance between two points in a square of side length 1 is the diagonal of the square. This can be calculated using the Pythagorean theorem, where the diagonal (d) is equal to the square root of the sum of the squares of the sides (a and b). In this case, the diagonal is equal to √(1^2 + 1^2) = √2 ≈ 1.41 units.

2. Can the maximum distance between two points in a square of side length 1 be greater than the diagonal?

No, the maximum distance between two points in a square of side length 1 cannot be greater than the diagonal. The diagonal represents the longest possible distance between any two points in a square with equal sides.

3. How does the maximum distance in a square of side length 1 compare to other geometric shapes?

The maximum distance in a square of side length 1 is the same as the diameter of a circle with a radius of 1 unit. This is also the longest distance between any two points in a regular hexagon or equilateral triangle with side length 1 unit.

4. Does the maximum distance change if the square is rotated?

No, the maximum distance between two points in a square of side length 1 remains the same regardless of rotation. This is because the diagonal remains constant regardless of the orientation of the square.

5. Is the maximum distance between two points in a square of side length 1 the same as the perimeter of the square?

No, the maximum distance between two points in a square of side length 1 is not the same as the perimeter of the square. The perimeter is the total distance around the square, while the maximum distance is the longest distance between any two points within the square.

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