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Math Help - Area of Region

  1. #1
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    Area of Region

    On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is contructed. The interiors of the square and the 12 triangles have no points in common. Let R be the region formed by the union of the square and all the trianlges, and let S be the smallest convex polygon that contains R. What is the area of the region that is inside S but outside R?

    I tried drawing the square and triangles several times but I must be misunderstanding the problem. Could i get some help pls?

    Vicky.
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  2. #2
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    Hello, Vicky!

    On each side of a unit square, an equilateral triangle of side length 1 is constructed.
    On each new side of each equilateral triangle, another equilateral triangle of side length 1 is contructed.
    The interiors of the square and the 12 triangles have no points in common.
    Let R be the region formed by the union of the square and all the trianlges,
    and let S be the smallest convex polygon that contains R.
    What is the area of the region that is inside S but outside R?
    Code:
                         B
        *-------*-------o
         *:::::*:*:::::* \
      *   *:::*:::*:::*   o C
      |::* *:*:::::*:* *::|
      |:::::*-------o:::::|
      |::*::|      A|::*::|
      *:::::|       |:::::*
      |::*::|       |::*::|
      |:::::*-------*:::::|
      |::* *:*:::::*:* *::|
      *   *:::*:::*:::*   *
         *:::::*:*:::::*
        *-------*-------*

    The region R is comprised of a unit square and 12 equilateral triangles.

    The area of R is: . A_R \;=\;1 + 12\left(\tfrac{\sqrt{3}}{4}\right) \;=\;1 + 3\sqrt{3}

    \Delta ABC has two sides AB = AC = 1 and included angle \angle BAC - 30^o
    Its area is: . \tfrac{1}{2}(1^2)\sin30^o \:=\:\tfrac{1}{4}

    The area of S is the area of R plus four of those triangles.
    . . A_S \;=\;(1 + 3\sqrt{3}) + 4\left(\tfrac{1}{4}\right) \:=\:2 + 3\sqrt{3}

    The area inside S and outside R is: . (2+3\sqrt{3}) - (1 + 3\sqrt{3}) \;=\;1

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, Vicky!

    Code:
                         B
        *-------*-------o
         *:::::*:*:::::* \
      *   *:::*:::*:::*   o C
      |::* *:*:::::*:* *::|
      |:::::*-------o:::::|
      |::*::|      A|::*::|
      *:::::|       |:::::*
      |::*::|       |::*::|
      |:::::*-------*:::::|
      |::* *:*:::::*:* *::|
      *   *:::*:::*:::*   *
         *:::::*:*:::::*
        *-------*-------*
    The region R is comprised of a unit square and 12 equilateral triangles.

    The area of R is: . A_R \;=\;1 + 12\left(\tfrac{\sqrt{3}}{4}\right) \;=\;1 + 3\sqrt{3}

    \Delta ABC has two sides AB = AC = 1 and included angle \angle BAC - 30^o
    Its area is: . \tfrac{1}{2}(1^2)\sin30^o \:=\:\tfrac{1}{4}

    The area of S is the area of R plus four of those triangles.
    . . A_S \;=\;(1 + 3\sqrt{3}) + 4\left(\tfrac{1}{4}\right) \:=\:2 + 3\sqrt{3}

    The area inside S and outside R is: . (2+3\sqrt{3}) - (1 + 3\sqrt{3}) \;=\;1
    Thanks!!!!!!

    Now I clearly understand what I was doing wrong.

    Vicky.
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