...so the teacher says.
Three equal circles with radius r are drawn as shown, each with its centre on the circumference of the other two circles. A, B and C are the centres of the three circles. Prove that an expression for the area of the shaded region is:
A = r^2 / 2 (Pi - _/3)
I couldn't even start to attack the question
Forgot to send an image of the circle....
I don't see the image "drawn as shown". I know what it looks like, but do not know what the shaded region is.
What is: "(Pi - _/3)"???
Last edited by nsl22 (2007-08-16 10:07:35)
you could probably use this:
and this: http://www.mathopenref.com/arcsector.html
to come up with an answer.
Consider the circle with centre A. The area of the sector bounded by the radii AB and AC and the arc BC (red area in diagram below) is.
This is also the area of the sector bounded by the radii BA and BC and the arc CA in the circle with centre B (blue area in diagram below) and that of the sector bounded by radii CB and CA and arc AB in circle with centre C (green area in diagram below).
Now add up all these areas. You get. But this actually counts the area of triangle ABC three times. So we must subtract twice the area of the equilateral triangle ABC, which is .
Hence the area of the shaded region is.
Last edited by JaneFairfax (2007-08-16 12:04:23)
Ive made three diagrams to illustrate what Im saying above.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
Hi, could someone explain why the area of each is
Last edited by jesjesjesjes (2014-02-08 07:13:37)
Welcome to the forum.
The circles have equal radii, so the triangle formed by the three circle intersection points is equilateral, ie. has angles of 60.
The area of a whole circle would be pi r squared, and a 60 sector is 1/6 of this.
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