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Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$.  Also, all the interior angles of the hexagon are equal.  Find the area of hexagon $IJKLMN$.

 Dec 18, 2023
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Here's how to find the area of hexagon IJKLMN:

 

1. Analyze the hexagon:

 

Since all sides have length 3 and all interior angles are equal, this hexagon is a regular hexagon.

 

2. Divide into equilateral triangles:

 

We can divide the hexagon into 6 congruent equilateral triangles by drawing diagonals from vertex I to the midpoints of the opposite sides. Each triangle has side length 3.

 

3. Calculate triangle area:

 

The area of each equilateral triangle can be calculated using the formula:

 

A_triangle = √3 / 4 * s^2

 

where s is the side length (3 in this case).

 

A_triangle = √3 / 4 * 3^2 = 9√3 / 4 square units

 

4. Calculate hexagon area:

 

The area of the hexagon is the sum of the areas of the 6 equilateral triangles:

 

A_hexagon = 6 * A_triangle

 

A_hexagon = 6 * 9√3 / 4 square units

 

5. Simplify:

 

A_hexagon = 27√3 / 2 square units

 Dec 18, 2023

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