Area of circle inscribed with 3 smaller circles

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Homework Help Overview

The problem involves a large circle that is inscribed with three smaller circles, each tangent to the others. The task is to find the area of the largest circle given the radius of the smaller circles.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to show the work leading to the proposed answer and explore various ideas, including finding the area of a triangle formed by the centers of the smaller circles and determining the radius of the larger circle.

Discussion Status

The discussion is ongoing, with participants suggesting starting points for the solution, such as examining the properties of the triangle formed by the centers of the smaller circles. There is no explicit consensus on a method yet, but ideas are being shared to guide the exploration.

Contextual Notes

Participants note that the triangle connecting the centers of the smaller circles is equilateral, which may influence the approach to finding the radius of the larger circle.

rayrenz
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Homework Statement


A large circle is inscribed with 3 smaller circles, eachhttps://www.physicsforums.com/newthread.php?do=newthread&f=156 of the four circles is tangent to the other three. If the radius of each of the smaller circles is a, find the area of the largest circle.

Homework Equations





The Attempt at a Solution


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the answer must be 1/3pi(7+4√3)a²
 
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Hi rayrenz - is there a question here ?

if its just to check you work, then you need to show how you got to the answer
 
ok, so what ideas do you have?
 
that's the problem to this, to show the solution on how to arrive the answer with r=a
 
maybe find the area of the triangle that connects the center of the three circles
 
not the area, but that triangle is a good place to start... at the end of the day, you want to find the radius R of the large circle

now back to that triangle... it will be an equilateral triangle - what is the length of one side?
 
then find the distance form the point of a triangle to both the large circle centre and edge, thus giving R
 

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