In hexagon ABCDEF, AB = DE = 2, BC = EF = 4, CD = FA = 6, and all the interior angles are equal. Find the area of hexagon ABCDEF.
Draw diagonals AC and DF. The hexagon is now divided into a parallelogram (rectangle?) and two congruent triangles.
AC and DF can be found using the law of cosines, since all the interior angles are 120 degrees.
To find the area of the hexagon ABCDEF, we can divide it into six congruent triangles. Each triangle has an equal base and height.
Step 1: Find the base and height of one of the triangles:
Since AB = DE = 2, the base of one triangle is 2.
To find the height, we can use the Pythagorean theorem.
Since BC = EF = 4 and CD = FA = 6, we can consider the triangle BCD. The base is BC = 4, and the hypotenuse is CD = 6.
Using the Pythagorean theorem, the height of the triangle BCD can be found as follows:
height^2 + base^2 = hypotenuse^2
height^2 + 4^2 = 6^2
height^2 + 16 = 36
height^2 = 36 - 16
height^2 = 20
height = √20 = 2√5
So, the height of one triangle is 2√5.
Step 2: Calculate the area of one triangle:
We have the base as 2 and the height as 2√5. Using the formula for the area of a triangle, A = 1/2 * base * height, we can find the area of one triangle.
Area of one triangle = 1/2 * 2 * 2√5 = 2√5
Step 3: Calculate the area of the hexagon:
Since the hexagon is composed of six congruent triangles, we can find the area of the hexagon by multiplying the area of one triangle by 6.
Area of hexagon ABCDEF = 6 * (Area of one triangle) = 6 * (2√5) = 12√5
Therefore, the area of hexagon ABCDEF is 12√5 square units.