Circle

Geometry Level 2

Consider a circle of radius R.
There is a square inscribed in this circle.
Another circle is inscribed in this square.
Another square is then inscribed in this inner circle.
Finally a circle is inscribed in this inner square.
There are 3 circles & 2 squares as shown in the figure.
What is the radius of innermost circle?

R/sqrt2 none of these sqrt2 * R R/2

Archiet Dev by Archiet Dev , 20, India

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6 solutions

Moshiur Mission
Apr 11, 2014

1st Circle: R 1st Square: (a/2)^2 + (a/2)^2 = R^2 so a = root2 R 2nd Circle: Radius = 1/root2 * R 2nd Square: (b/2)^2 + (b/2)^2 = (1/root2 * R)^2 so b = R 3rd Circle Radius = R/2

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Can you make it clearer? what is "a" actually?

Hafizh Ahsan Permana - 7 years, 1 month ago

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a is the side of the square

Aneesh Cool - 7 years, 1 month ago

diameter is R/2 not radius and in que its asked radius not diameter

Arpan Soni - 7 years, 1 month ago

for every next interior circle r=Rcos(45)

Naveen Verma - 7 years, 1 month ago

Create a triangle with the help of the radius and one of the sides of the square. The triangle so created is a 45-45-90 triangle. With the help of this you get the radius of Second inner circle i.e. R/sqrt2 by repeating this same procedure again(once), you get R/2

Kshitij Dadhekar - 7 years, 1 month ago
Mahendar Singh
Apr 15, 2014

R be radius of outer circle; R1 radius of inner circle = (R/ sqrt(2)) R2 radius of inner most circle = (R/sqrt(2))*(1/sqrt(2)) = R/2

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John Mavely
Apr 17, 2014

R is diagonal of first square, hence side of square becomes R/root(2). Side of 1st square= radius of middle circle. Now again repeat the same thing(i.e;divide by root(2). You end up with R/2.

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Zarree Khan
Apr 16, 2014

right ans is r/2 ,it can b judge by appearance...

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Meran Arshad
Apr 15, 2014

Take a broader view of picture, and try to see the circles only

what you will find the inner most circle radius to be the half of the largest circle. hence you can conclude that

if the radius of largest circle is R then the radius of innermost circle will be half that is R/2.

I hope this will be helpful.

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Krishna Garg
Apr 15, 2014

Since ist inner square diagonalhas radius R,Cicle inscribed in this will have radius R under root 2 (Pythogorous theorem with issocellous triangle having diagonal R.) Similarly ,next inscribed circle will have radius R/2 Ans. K.K.GARG,India

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Did not understand ur solution!!!!!!!!!!!!!!!!!! PLZ explain

Debendra Kumar Sahu - 7 years, 1 month ago

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in the inscribed circle,you have square's diagonal.anow in this circle you can make a triangle with radiurs of circle perpendicular to each other and square's diameter as hypotenuse.taling R ( hypotenuse, and right angled triangle ,we can find radiur as R/under root 2.same way for next circle.If you need vfurther details please contact me on 098 290 38864. Thanks. K.K.GARG,India

Krishna Garg - 7 years, 1 month ago

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