Solve it if you dare

Geometry Level 3

Three circle with radii 3, 2, 1 touch each other making a gap in the middle, find the area of that gap.

Give your answer to 3 decimal places.

You may use a calculator.


The answer is 0.464.

View wiki
Ariful Hoque by Ariful Hoque , 23, Bangladesh

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Ronak Agarwal
Aug 2, 2014

Arcs Arcs

If we join the centres of the circles we will have a figure as shown.

Area of the gap = Area of the triangle-Area of the three arcs.

= ( 1 2 × 3 × 4 ) − ( 2 2 A 2 + 1 2 B 2 + 3 2 C 2 ) A l s o A = s i n − 1 ( 4 5 ) = 0.927 , B = s i n − 1 ( 1 ) = 1.57 , C = s i n − 1 ( 3 5 ) = 0.643 =(\frac { 1 }{ 2 } \times 3\times 4)-({ 2 }^{ 2 }\frac { A }{ 2 } +{ 1 }^{ 2 }\frac { B }{ 2 } +{ 3 }^{ 2 }\frac { C }{ 2 } )\\ Also\quad A={ sin }^{ -1 }(\frac { 4 }{ 5 } )=0.927,B={ sin }^{ -1 }(1)=1.57,C={ sin }^{ -1 }(\frac { 3 }{ 5 } )=0.643

Put the values to get Area of the gap= 0.464 0.464

6 Helpful 0 Interesting 0 Brilliant 0 Confused

This problem is not hard at all, but one has to make use of a calculator.

Dieuler Oliveira - 6 years, 10 months ago

Log in to reply

Agree with you. I wonder how such problems are highly rated

Ronak Agarwal - 6 years, 10 months ago

But its fun no? I mean realizing that 1,2,3 radius would lead to a right angled Triangle at the center :D.

Mithun K - 6 years, 10 months ago

:) I wish I thought of trig

genius kid - 1 year ago
Rifath Rahman
Sep 17, 2014

As sides are 3-4-5 so its right angle,then use (cos^-1) u get the ans with the help of calculator 0.464

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...