# Intermediate Geometry : How to find the length of the diagonal of a hexagon

## Example Questions

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### Example Question #21 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is , find the length of diagonal .

Explanation:

When the regular hexagon is divided into  congruent equilateral triangles, it's easy to see that diagonal  is comprised of two heights of two equilateral triangles. This holds true for the other  diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

Plug in the given perimeter to find the length of a side for the given hexagon.

Notice that the height of the equilateral triangle creates two congruent  triangles whose side lengths are in the ratio of .

Thus, we can set up the following proportion to find the length of the height:

Since the diagonal is made up of two of these heights, multiply by  to find the length of the diagonal.

### Example Question #22 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is , find the length of diagonal .

Explanation:

When the regular hexagon is divided into  congruent equilateral triangles, it's easy to see that diagonal  is comprised of two heights of two equilateral triangles. This holds true for the other  diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

Plug in the given perimeter to find the length of a side for the given hexagon.

Notice that the height of the equilateral triangle creates two congruent  triangles whose side lengths are in the ratio of .

Thus, we can set up the following proportion to find the length of the height:

Since the diagonal is made up of two of these heights, multiply by  to find the length of the diagonal.

### Example Question #23 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is , find the length of diagonal .

Explanation:

When the regular hexagon is divided into  congruent equilateral triangles, it's easy to see that diagonal  is comprised of two heights of two equilateral triangles. This holds true for the other  diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

Plug in the given perimeter to find the length of a side for the given hexagon.

Notice that the height of the equilateral triangle creates two congruent  triangles whose side lengths are in the ratio of .

Thus, we can set up the following proportion to find the length of the height:

Since the diagonal is made up of two of these heights, multiply by  to find the length of the diagonal.

### Example Question #24 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Explanation:

When the regular hexagon is divided into  congruent equilateral triangles, it's easy to see that diagonal  is comprised of two heights of two equilateral triangles. This holds true for the other  diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

Plug in the given perimeter to find the length of a side for the given hexagon.

Notice that the height of the equilateral triangle creates two congruent  triangles whose side lengths are in the ratio of .

Thus, we can set up the following proportion to find the length of the height:

Since the diagonal is made up of two of these heights, multiply by  to find the length of the diagonal.

### Example Question #25 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Explanation:

When the regular hexagon is divided into  congruent equilateral triangles, it's easy to see that diagonal  is comprised of two heights of two equilateral triangles. This holds true for the other  diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

Plug in the given perimeter to find the length of a side for the given hexagon.

Notice that the height of the equilateral triangle creates two congruent  triangles whose side lengths are in the ratio of .

Thus, we can set up the following proportion to find the length of the height:

Since the diagonal is made up of two of these heights, multiply by  to find the length of the diagonal.

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