6798 /www/nrich/html/content/id/6798/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <br></br> <p>The diagram show four circles each of which touches the largest square and two adjacent circles. A second square has its vertices at the midpoints of the sides of the largest square and the central square has its vertices at the centres of the circles.<br></br> <mdo:image alt="" height="135" src="Shaded%20Circle.jpg" width="138"></mdo:image><br></br> What is the ratio of the total shaded area to the area of the outer square?<br></br> <br></br> If you liked this problem, <a href="http://nrich.maths.org/public/viewer.php?obj_id=553&amp;part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas.</p> <p> </p> <p> </p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>The middle sized square passes through the centres of the four circles. Each side of the middle sized square together with the edges of the outer square creates a right angled isosceles triangle with angles of 45$^\circ$. Thus the angles these sides make with the inner square are also 45$^\circ$.</p> <p>Each side of the middle sized square bisects the area of the circle. The inner half of that circle is made up of two shaded segments with angles of 45$^\circ$ which together are equal in area to the unshaded right angled segment.</p> <p>Thus the total shaded area is exactly equal to the area of the inner square and hence equal to one quarter of the area of the outer square.</p> <p> </p></mdoxml> 2 3 0 0 1 0 0 Measures and Mensuration Area Fractions, Decimals, Percentages, Ratio and Proportion Calculating with ratio & proportion