Circle fun

Geometry Level 4

If all circles with centers A , B , C , D , E A, B, C, D, E and F F have the same radius of 7, what is the area of the shaded region given that circles E E and F F pass through the centers of the other circles and that ABCD forms a rectangle (when joining up those points with lines).

Give answers to at least one decimal place.​

Harder problem .


The answer is 24.680374339.

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Samy Adams by Samy Adams , 23, United Kingdom

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4 solutions

Samy Adams
Jun 9, 2016

All angles are 60 degrees, as the following lines cut the circle into 6 equal parts, hence the angle is 360/6 = 60 deg,

Therefore,

Area of Shaded Region = The two triangles - Half of the four "lips"

Area of one "half lip":

The shape of a segment arises, and so, using the formula for area of a segment...

Angle = 60 deg = π 3 \frac{π}{3} radians

Therefore,

Half the area of one lip = ( 7 2 2 (\frac{7^2}{2} )*( ( π 3 \frac{π}{3} ) - sin( π 3 \frac{π}{3} ) )

Area of one triangle(Using pythagorean theorem) = ( 1 2 (\frac{1}{2} x 7 x sqrt{ 7 2 − 3. 5 2 7^2 - 3.5^2 } )

So, Area of shaded region = (2 x triangle areas) - (4 x "half lip" areas) = 24.68037...

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Sidenote: The 60 degrees also arises from the formation of equilateral triangles which arise from the fact that the circles have equal radii.

Samy Adams - 5 years ago

"All angles are 60 degrees, as the following lines cut the circle into 6 equal parts, hence the angle is 360/6 = 60 deg," is bad reasoning. I wrote this a while ago so apologies. I believe that using the fact that circles have equal radii by joining up lines of equal length 7 create the equilateral triangles proving that the angle involved in the circle segment calculation is 60 degrees.

Samy Adams - 11 months, 2 weeks ago

In my opinion the problem was explained ambiguously. It is not enough to say that circles E and F pass through the centres of the other circles. The idea is complete if you add two additional circles (e.g. G G , H H ) with the same assumptions (the circles pass through the centers of the other circles). See the figure below. The answer is:

2 3 r 2 4 − 4 r 2 12 ( 2 π − 3 3 ) = 24.680374339 2\frac{\sqrt{3}r^2}{4} - 4\frac{r^2}{12}(2\pi - 3 \sqrt{3}) = 24.680374339

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You're right. I've updated the description to say that ABCD forms a rectangle.

Samy Adams - 11 months, 2 weeks ago
Michael Mendrin
Jun 9, 2016

I think you should at least point out that circles E E and F F passes through the centers of the other 4 4 circles. I missed that detail the first time I tackled this. Once I saw that missed detail, the rest more or less followed.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

oops! Thanks, ill update it now.

Samy Adams - 5 years ago

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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