The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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 

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 

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 

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   
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.