square PQRS

tony123 · 2007-12-01 05:02:47

Inside the unique square PQRS are drawn the 4 quarter-circles having unit radii and the vertices as centers. Determine the area A that is common to the 4 quadrants.

JaneFairfax · 2007-12-01 06:53:23

The area of the square is

The area of each quadrant is

Now let T be the instersection of the quarter-circle with centre Q and the quarter-circle with centre R. Then the area of the equilateral triangle TQR is

. The area of the sector bounded by RQ, RT and the arc QTS (call it ) is . This is also the area bounded by QR, QT and the arc RTP (call it ). Hence

Thus we have three simulataneous equations in three unknowns. So, solve them! Whee!

Last edited by JaneFairfax (2007-12-01 08:32:55)

JaneFairfax · 2007-12-01 07:35:16

I got

If you wanna check, here are my answers for B and C as well.

gyanshrestha · 2007-12-02 23:37:40

hi JaneFairFax
i amnot understanding how we can conclude that .

Please can u give me its reasons.

Last edited by gyanshrestha (2007-12-02 23:38:40)

mathsyperson · 2007-12-03 00:26:02

The triangle RQT is equilateral, because each of its sides is a radius.
Therefore, each of its angles is 60°.

That means that the sector A_1 has an angle of 60°, and so its area is 1/6 of the whole circle's area, which is π/6.

JaneFairfax · 2007-12-03 00:50:52

gyanshrestha wrote:

hi JaneFairFax
i amnot understanding how we can conclude that .

Please can u give me its reasons.

A sector formed in a circle of radius r by two radii with angle θ between them has area

. (Note that θ must be in radians in the formula.)

Math Is Fun Forum

#1 2007-12-01 05:02:47

square PQRS

#2 2007-12-01 06:53:23

Re: square PQRS

#3 2007-12-01 07:35:16

Re: square PQRS

#4 2007-12-02 23:37:40

Re: square PQRS

#5 2007-12-03 00:26:02

Re: square PQRS

#6 2007-12-03 00:50:52

Re: square PQRS

Board footer