School-Safe Puzzle Games

How Much Land, A Geometry Puzzle

How much land can be enclosed by a dozen 16 foot beams?

update: yes, the beams are straight.

Answers can be submitted below, will unmask Wednesday or Thursday

10 Comments to “How Much Land, A Geometry Puzzle”

  1. joe | Profile

    As we are not told if the beams are straight or not, then by having the perfect curvature to form a perfect circle we will have the maximum area. The circumference would be (12×16=) 192 feet (using circumeference = 2 x (pi) x radius).
    This would give a radius of 30.5578 feet.
    The area (pi x r-squared) would be 2,933.55 feet squared.
    (If the beams are straight then the maximum area is a regular dodecagon of side 16 feet. This area would be 2,866.21 feet sq.)

  2. Shawn | PUZZLE GRANDMASTER | Profile

    2,866.215 ft^2, if arranged into a 12-sided figure

  3. Hex | PUZZLE MASTER | Profile

    I guess a 12 sided polygon should yield the biggest area.
    A = 12 x L x D / 2
    tan alpha = L / 2 / d
    A = 3 x L^2 / tan(360/24) = 2,866.22 sq ft

  4. Falwan | Profile

    Area of a dodecagon (12 sided ) = 11.19615242 * a * a = 11.19615242 (16)(16) = 2866.21501952 sq. ft.

  5. Bobo The Bear | PUZZLE MASTER | Profile

    The greatest area would be enclosed by a regular dodecagon.
    The area would be 2866.215 square feet.

  6. brianu | Profile

    2967.32 square feet

  7. cheddarmelt | Profile

    for regular polygons, area = {(S^2)N}/{4 tan (pi/N)}, where S=side length and N=number of sides, and tangent is in radians.

    I came up with 2866 sq ft.

  8. Hendy | Profile

    First cheat and go to

    Second enter the side length.

    Third do the calculations.

    Four post the answer. A = 2866.2150202129

  9. ihavenoorgans | Profile

    Use three beams to make a triangle around yourself and declare yourself to be on the outside. You have now enclosed the entire planet.

  10. Tsopi | Profile

    Well i don’t really know what a beam is becouse i am not english or american but if we have a dozen of them (12) and each one is 16 feet and as you say they are straight things…we can make a square with a side of a = 3 beams = 3 *16 = 48 foot. to calculate the land it includes we multiply 48 * 48 = 2304…but this doesn’t match with the other answers…what am i missing?

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