PSAT Math help » Geometry » airplane Geometry » Hexagons » how to discover the length of the diagonal line of a hexagon     Explanation:

A diagonal is a heat segment joining two non-adjacent vertices the a polygon. A consistent hexagon has six sides and also six vertices. One vertex has three diagonals, for this reason a hexagon would have three diagonals times 6 vertices, or 18 diagonals. Division this number through 2 come account for duplicate diagonals in between two vertices. The formula because that the variety of vertices in a polygon is: where .

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Explanation:

A diagonal line connects two non-consecutive vertices the a polygon. A hexagon has six sides. There room 3 diagonals native a single vertex, and also there are 6 vertices on a hexagon, which suggests there would certainly be 18 diagonals in a hexagon. However, we need to divide by 2 as fifty percent of the diagonals are typical to the exact same vertices. For this reason there room 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is: where n = the variety of sides in the polygon.

Thus a pentagon thas 5 diagonals. An octagon has 20 diagonals.

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### Example question #1 : Hexagons

Regular Hexagon  has a diagonal with length 1.

Give the size of diagonal  Explanation:

The an essential is come examine  in thie figure below:

Each inner angle that a regular hexagon, including , measures , so it deserve to be conveniently deduced by means of the Isosceles Triangle organize that   .

Also, through symmetry, ,

so ,

and  is a  triangle whose long leg has length .

By the  Theorem, , i beg your pardon is the hypotenuse of , has actually length times the of the long leg, so .

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### Example inquiry #1 : just how To discover The size Of The diagonal Of A Hexagon

Regular Hexagon  has a diagonal with length 1.

Give the size of diagonal  Explanation:

The an essential is come examine  in thie figure below:

Each internal angle the a constant hexagon, including , measures , therefore it deserve to be easily deduced by means of the Isosceles Triangle organize that   .

Also, through symmetry, ,

so ,

and  is a  triangle who hypotenuse has length .

By the  Theorem, the lengthy leg of  has length times the of hypotenuse , so .

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### Example concern #3 : Hexagons

Regular hexagon  has side size of 1.

Give the length of diagonal  Explanation:

The key is come examine  in thie number below:

Each internal angle the a constant hexagon, consisting of , measures , therefore it have the right to be conveniently deduced by method of the Isosceles Triangle theorem that . To find we can subtract  from . Thus resulting in: Also, through symmetry, ,

so .

Therefore, is a  triangle whose brief leg has size .

The hypotenuse of this  triangle steps twice the size of short leg , so .

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### Example concern #4 : Hexagons

Regular hexagon  has side length 1.

Give the size of diagonal  Explanation:

The crucial is come examine  in thie figure below:

Each internal angle the a consistent hexagon, including , measures , for this reason it have the right to be conveniently deduced by means of the Isosceles Triangle theorem that . To find we subtract from . Therefore resullting in Also, through symmetry, ,

so ,

and is a  triangle whose short leg has length .

See more: How Many Slices Does A Large Pizza Have, How Many Slices Are In A Large Pizza

The lengthy leg of this  triangle steps times the size of quick leg , so .

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