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# Hexagon

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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