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: _____
To solve for the area of a hexagon with sides equal to 4 cm, we can follow these steps:
Step 1: How many equilateral triangles are there?
In a regular hexagon, each of the six sides is equal in length. Since a hexagon can be split into six equilateral triangles, there are 6 equilateral triangles in total.
Step 2: If we cut an equilateral triangle down the middle (using the green line), what special triangle do you create?
Cutting an equilateral triangle down the middle creates two congruent 30-60-90 triangles.
Step 3: Sketch and label all side lengths of the 30-60-90 triangle by itself below.
In a 30-60-90 triangle, the ratio of the side lengths is 1 : √3 : 2. We can label the shortest side as "x," the side opposite the 60-degree angle as "x√3," and the hypotenuse as "2x."
So, in the 30-60-90 triangle:
- Short side (x)
- Side opposite the 60-degree angle (x√3)
- Hypotenuse (2x)
Step 4: What is the length of the short side of one 30-60-90 triangle?
The length of the short side (x) is equal to 4 cm since the sides of the hexagon are said to be 4 cm.
Step 5: What is the length of the hypotenuse of one 30-60-90 triangle?
The length of the hypotenuse (2x) is 2 times the length of the short side. So, the hypotenuse of one 30-60-90 triangle is (2 * 4 cm) = 8 cm.
Step 6: Apply properties of 30-60-90 triangles to calculate the long leg:
In a 30-60-90 triangle, the length of the long leg (side opposite the 30-degree angle) is √3 times the length of the short side. So, the long leg is (√3 * 4 cm) = 4√3 cm.
Step 7: What is the vocabulary word for the long side of the 30-60-90 called in the polygon (green line)?
The vocabulary word for the long side of the 30-60-90 triangle within the hexagon is the "apothem."
Step 8: What is the height of the equilateral triangle?
The height of an equilateral triangle is the length of the apothem, which we found to be 4√3 cm.
Step 9: Apply the triangle area formula to calculate the area of one equilateral triangle:
The area of an equilateral triangle can be calculated using the formula A = (sqrt(3) / 4) * (side length)^2. Plugging in the side length of the hexagon, which is 4 cm, we get:
A = (sqrt(3) / 4) * (4 cm)^2
= (sqrt(3) / 4) * 16 cm^2
= 4√3 cm^2.
Step 10: Calculate the area of the complete hexagon by multiplying the area of one equilateral triangle by the number of triangles:
Since there are 6 equilateral triangles in the hexagon, multiplying the area of one triangle (4√3 cm^2) by 6 gives us the area of the complete hexagon.
Area of the hexagon = 6 * 4√3 cm^2
= 24√3 cm^2.
Therefore, the area of the hexagon with sides equal to 4 cm is 24√3 cm^2.