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HW # 12
Octagoning the Circle

 

Same as hexagoning the circle, but using 8 sides and 8 triangles instead of 6.

 

 

 

 

 
 
 
 
 
 

 
THE PROBLEM
1. If the radius = 10 units, find the perimeter  and area of the circumscribed hexagon
 
2. Generalize your work in Question 1 to a circle of radius r
 
 
 
 
Things that you need to remember:
Area of a triangle = (b * h) ÷ 2
 
Sine of A = Opposite Side ÷ Hypotenuse
Cosine A = Adjacent Side ÷ Hypotenuse
Tangent A = Opposite Side ÷ Adjacent Side
 
QUESTION 1
You may recall finding the perimeter and the area of a hexagon in the unit Do bees Build It Best? from Year 2 by subdividing the hexagon into central equilateral triangles.
 
Well, that is exactly what you should do.
Lets start by isolating one of the equilateral triangles shown here.
 
Now, since the complete circle is 360 o
the measurement of the angle A is 360 ÷ the number of triangles.
 
How does the base of this triangle compare with the sides of the hexagon?
 
Great!
Now label every part of the triangle.
 
Remember, the radius is equal to 10 units.
Note: The known values are represented in red.:
 
 
 
 
 
 

 

                              A

 

              

                            h  = r  

 

                            S

 

 
 
 
 
 
Do you see any right triangle within the equilateral triangle?
Can you isolate the right triangle?
 
Okay, draw the right triangle and label the new angle and the new base.
Did you notice that the new angle is a half of the angle A?
Did you notice that the new base is a half of the base S?
Make sure that your new triangle reflects that change.
 
By now you should have a right triangle with the height and an angle known. Also, you know that the base is half of one of the sides of the hexagon.
 
Why are you doing all this? Think...
If you don't know click here
 
Your right triangle should look something like this.
Of course you will use the actual value for the
angle = A/2 and for the height of the triangle.
 
Now, using the tangent find the value of x.
 
Tan (A/2) = x/h  and solving for x.
  
   x = h * Tan (A/2)
 
Use your calculator to find the value of x
 
 
 
 
 
 
 
 
 
 
 
           A/2
 
 
 
        h
 
 
 
 
                x=  S/2
 
 
Now find the value of S.
S = 2x      (Can you explain why?)
 
 
Since you now know the length of a side of the regular hexagon you can find the area of one of the triangles.
 
A = b * h
          2
                              Where  b = s
 
 
Substitute the values and find the area of the triangle.
 
Now find the area of the entire hexagon.
 
How many triangles are there?
 
Area of the Hexagon = (# of triangles )*( Area of one triangle).
 
You can also find the perimeter of the regular hexagon.
 
Perimeter of the Hexagon = (# of Sides)*(Length of a Side)
 
 
QUESTION 2
 
Follow the same steps using h =r instead of 10. Make sure to reduce every term.
For instance,  x = r * Tan (A/2)
 
Remember that generalize means "for any radius r"
 
You should get something like this:
 
 
     Shape          Perimeter      Area
Circle     kcr = 6r   ka= 3r2
Square 8r 4r2
Regular Hexagon 6.928r 3.4641r2
Regular Octagon 6.63r 3.31r2
 
 
Now you should be ready to answer questions like these:
 
What happens to kc and ka as the number of sides increases?
Why?
 
What is the relationship between ka and kc?
Why?
 
Or     ka   = ?
        kc?
 
Great!
Now get ready to share your findings with your class.
 
 
Mr. Lora

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