Arcs Inside a Square

Date: 07/25/99 at 20:29:57
From: Susan
Subject: Geometry and problem solving

I can't draw this problem, so I'll explain it to you. Draw a square 
ABCD. Two arcs are drawn between points A and C (inside the square). 
Two arcs are drawn between points B and D (inside the square). If each 
side of the square is 5, what is area of the square-like figure (it is 
not a square) that is created by the intersection of the arcs in the 
square?

Date: 07/26/99 at 14:58:19
From: Doctor Rick
Subject: Re: Geometry and problem solving

Hi, Susan.

I think I have the correct figure. You didn't say where the center of 
each arc is, but I'm guessing that the 2 arcs from A to C have centers 
at B and D, and the 2 arcs from B to D have centers at A and C. Is 
this correct?

Label the intersections of the arcs E, F, G, and H, so that the 4 arcs 
are AEFC, BFGD, CGHA, and DHEB. Put point J at the center of the 
square.

Divide the region EFGH into 4 equal parts by drawing lines EG and FH. 
All you need to find now is the area of one of the 4 parts, such as 
EJF.

Draw lines DE, DF, and DJ. 

   

You can find the area of EJF by subtracting the areas of 2 congruent 
triangles from the area of a sector of a circle. To do this, you will 
need to find the angle EDF and the length JF. See what you can do.

There may be other ways to do it, but this will work. Have fun!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/