# Intermediate Geometry : Plane Geometry

## Example Questions

### Example Question #151 : Hexagons

In the regular hexagon above, if the length of diagonal  is , what is the area of the hexagon?

Explanation:

When all the diagonals of a hexagon are drawn in, you should notice that the long diagonals, marked by solid lines, form  congruent equilateral triangles. You should also notice that the other diagonals drawn in with dashed lines are also the heights of two of the equilateral triangles.

Start by using the diagonal to find the length of a side of the hexagon.

Cut the diagonal in half so that we are just left with the height of one equilateral triangle. Notice that the height cleaves the equilateral triangle into two congruent  triangles whose sides are in the ratios of .

Set up a proportion to solve for the length of a side of the triangle.

Plug in the given diagonal to solve for the side length.

Now, recall how to find the area of a regular hexagon:

Plug in the side length that you just found in order to find the area.

Make sure to round to  places after the decimal.

### Example Question #152 : Hexagons

In the regular hexagon above, if diagonal  has a length of , what is the area of the hexagon?

Explanation:

When all the diagonals of a hexagon are drawn in, you should notice that the long diagonals, marked by solid lines, form  congruent equilateral triangles. You should also notice that the other diagonals drawn in with dashed lines are also the heights of two of the equilateral triangles.

Start by using the diagonal to find the length of a side of the hexagon.

Cut the diagonal in half so that we are just left with the height of one equilateral triangle. Notice that the height cleaves the equilateral triangle into two congruent  triangles whose sides are in the ratios of .

Set up a proportion to solve for the length of a side of the triangle.

Plug in the given diagonal to solve for the side length.

Now, recall how to find the area of a regular hexagon:

Plug in the side length that you just found in order to find the area.

Make sure to round to  places after the decimal.