Question 1199598: Ms.Ikelyn, for the solution you gave for the problem here:
https://www.algebra.com/tutors/students/your-answer.mpl?question=1199595
Doesn't make sense to me. Because if we subtract the area of the large circle and the small circles, the difference is more than just the "a" regions. Please let me know how to overcome this obstacle. Thanks!
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
We have this original diagram,
which is a color-coded version of the diagram you posted.
The stuff in blue is what we want, which is all of the regions marked "a".
The template calculation is this
A-B-C
where,
A = area of the largest circle
B = total area of the 7 smaller circles added up
C = area of all the green regions added up
The values of A and B should be fairly straight forward to calculate.
area of a circle = pi*r^2
r = radius
I'll let the student do this part.
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The issue is: what do we do to calculate the area of the green regions?
This is one of those problems, like many topics in math, where it helps to try to break it up into smaller bite-sized pieces.
The key is to break it up into pieces you are more familiar with.
Also, some trial and error (and a bit of luck) is often involved.
With more practice, you should be able to spot the patterns easier so you can figure out how to break the figure up.
Let's form a regular hexagon as shown below.
With exception of the middle-most circle, each of the other circle centers form vertices of this hexagon.
Notice how a regular hexagon is really a combination of 6 equilateral triangles.
Highlight one of those triangles
Then zoom in further:
to see that the equilateral triangle consists of 4 key ingredients:- 3 pizza slices (except they are "inverted", so to speak, rather than arranged into a circle). These pizza slices are shown in white.
- The fourth ingredient is the inner green region.
Now why are we doing all this? Well the equation to find the area of a equilateral triangle is
A = 0.25*s^2*sqrt(3)
where 's' is the side length of the equilateral triangle.
In this case, s = 2r = 2*3 = 6 cm
The area of that blue equilateral triangle is
A = 0.25*s^2*sqrt(3)
A = 0.25*6^2*sqrt(3)
A = 9*sqrt(3)
Now we must subtract off those white pizza slices that form part of the blue triangle.
Those 3 white slices can be rearranged to form a semicircle
Quick Reasoning: each slice is 60 degrees, three of them give 3*60 = 180 degrees out of 360 total. This is 180/360 = 1/2 of a full pizza.
area of a semicircle = 0.5*(area of a full circle)
area of a semicircle = 0.5*pi*r^2
area of a semicircle = 0.5*pi*3^2
area of a semicircle = 4.5pi
Therefore,
area of green region = (area of equilateral triangle) - (area of semicircle)
area of green region = 9*sqrt(3) - 4.5pi
This is the green region mentioned in this diagram
specifically the green region contained inside the blue equilateral triangle.
But if you refer back to this diagram
it's clear we have 6 identical green regions. So you'll have to multiply the expression 9*sqrt(3) - 4.5pi by 6.
I'll let the student do this.
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Summary:
The final answer is of the form
A-B-C
A = area of the largest circle
B = total area of the 7 smaller circles added up
C = area of all the green regions added up
I'll let the student fill in the numeric details, and simplify if needed.
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