Question

Score for Question 3: ___ of 5 points)

3. Use the diagram of a REGULAR HEXAGON and follow these steps to solve for the area of a hexagon with sides equal to 4 cm. Leave radicals in your answers.

How many equilateral triangles are there? __________
If we cut an equilateral down the middle (green line) what special triangle do you create? __________
Sketch and label all side lengths of the 30-60-90 triangle by itself below.
What is the length of the short side of one 30-60-90 triangle? __________
What is the length of the hypotenuse of one 30-60-90 triangle? __________
Apply properties of 30-60-90 triangles to calculate the long leg: ¬¬¬¬___________
What is the vocabulary word for the long side of the 30-60-90 called in the polygon (green line)? ____________
What is the height of the equilateral triangle? _________
Apply triangle area formula to calculate the area of one equilateral triangle: ____________
Calculate the area of the complete hexagon by multiplying area of one equilateral triangle by # of triangles:

Answer 1

(a)6 Equilateral Triangles.

(b)30-60-90 triangle.

(c)See Attached Triangle in Figure 2

(d)Length of the short leg = 2cm.

(e)Length of the hypotenuse= 4cm.

(f)Length of the long leg=
`2√(3)cm`

(g)Apothem.

(h)Height of the Equilateral triangle=
`2√(3)cm`

(i)Area of One Equilateral triangle =
`4√(3)cm^2`

(j)Area of Hexagon =
`=24√(3)\:cm^2`

Explanation:

(a)From Figure 1, there are 6 Equilateral Triangles.

(b)If we cut an equilateral down the middle (green line), we create a 30-60-90 triangle.

(c)Triangle Attached in Figure 2.

(d)The length of the short leg of one of the 30-60-90 triangle is 2cm.

(e)The length of the hypotenuse of one of the 30-60-90 triangle is 4cm.

(f)Length of the long leg

We use Pythagoras Theorem to find the length of the long leg of the right triangle.

`4^2=2^2+l^2\\l^2=4^2-2^2=12\\l=√(12)=2√(3)cm`

(g)The vocabulary word for the long side of the 30-60-90 called in the polygon (green line) is Apothem.

It is line segment from the center to the midpoint of one of the sides of a polygon.

(h)Height of the Equilateral Triangle=
`2√(3)cm`

(i)Area of One Equilateral triangle

Base =4 cm, Height =
`2√(3)cm`

Area=0.5X4X
`2√(3)cm`

=
`4√(3)\:cm^2`

(j)Area of Hexagon =Area of One Equilateral Triangle X 6

`=6X 4√(3)\\=24√(3)\:cm^2`