There are three tangent circles that all have a radius of 1. What is the perimeter of the space in between?
September 9th, 2010 | 
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This is the closest diagram I can find: http://cairnarvon.rotahall.org/pics/descartes2.png (ignore the numbers there and just look at the shapes)
If each of those circles is equal and has a radius of 1, what is the perimeter of that little space right in the middle of them?
This is a GRE question and I have no idea how to answer it, so a step-by-step guide would be very helpful! Thanks!
The book is giving the answer as pi.
I think I understand. The area of the triangle inside is sqrt(3), but the area of the remaining parts that are within the triangle form a circle that is half the size of the other circles, so the diameter is 1, which would make the circumference pi(1), or pi.
Since the perimeter of the area in between is made up of pieces of the inner circle (albeit not in order), it is equal to the circumference.
Does that sound about right?
Tricky tricky!
Posted in gre guide | 
Tags: circles, circumference, diameter, form a circle, gre, inner circle, little space, perimeter, pi, png, radius, shapes, sqrt, triangle
Three 60 degree pieces of arc equals one 180 degree piece of arc, equals half the circumference of one circle. C/2 = (2 pi r) / 2 = pi r, but r= 1, so the answer is just pi.
Edit: nevermind! explanation for perimeter of the little piece is more than adequately explained above and below! This was an object lesson on how important it is to read the question and make sure you’re answering the right question!
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Connect the centers of the circles to create an equilateral triangle with sides of length 2
The area of this triangle is (1/2)(2)sqrt(3) = sqrt(3)
you can use a 30-60-90 triangle to find the altitude of this triangle (sides in ratio of 1:sqrt(3):2)
the triangle intersects each circle with the same sector area
area of sector = pi*r^2 / 6 (since the central angle is 60 degrees, or 1/6 of the entire circle)
area of each sector = pi / 6 (radius = 1), so 3 of them have a combined area of 3pi/6 = pi/2
difference between area of triangle and area of sectors will be area of the "little space":
sqrt(3) – pi/2
if this is a multiple choice question, this would probably be the answer choice
if not, or it’s a grid in, this has a numeric value of about .161 sq. units
Note: this technique is fairly common in standardized tests… that is, using the difference of two areas to find an odd shaped area (like the little piece in the center of the circles).
Another example would be two tangent circles inscribed in a rectangle, and finding the area outside the circles, but inside the rectangle. This would be area of rectangle – area of circles. Again, an area you can’t calculate directly…