How many 1-unit equilateral triangles does a hexagon with side-3 units contain?

Formula for the area of an equilateral triangle with side length

a

is

A

t

=

√

3

4

⋅

a

2

Let

2

x

be the side length of the equilateral triangle,

given that area of the equilateral

A

t

=

2

units

2

⇒

A

t

=

√

3

4

⋅

(

2

x

)

2

=

√

3

4

⋅

4

x

2

=

2

⇒

x

2

=

2

√

3

units

2

A regular hexagon can be divided into 6 congruent equilateral triangles, as shown in the figure.

given that the equilateral triangle and the regular hexagon have equal perimeter,

⇒

side length of the hexagon

=

3

⋅

2

x

6

=

x

units

⇒

area of the regular hexagon

=

A

h

=

6

⋅

√

3

4

⋅

x

2

⇒

A

h

=

6

⋅

√

3

4

⋅

2

√

3

=

3

units

2