Rolling Around

2159 /www/nrich/html/content/04/02/penta4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Describe the locus of the centre of the circle and its length. <a href="/content/04/02/penta4/roundabout.swf">Full screen version</a> <mdo:flash height="400" id="/content/04/02/penta4/roundabout.swf" width="400"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/04/02/penta4/roundabout.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="400"></param> </mdo:flash> If the circle now rolls around an equilateral triangle, can you describe the locus of the centre of the circle and its length? Can you generalise your findings? Here are two related problems you might like to take a look at: <a href="/2162">Rollin' Rollin' Rollin'</a> <a href="/492">Is There a Theorem?</a> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Faisal from Arnold House School offered a strategy for working on this problem: What we did was take a dart board and rolled it around 4 metre rulers. It travelled a straight line along the ruler. The bullseye of the board stayed the same distance away from the ruler. After each corner the bullseye made a curve a quarter of a circle (which is 90 degrees) so after the experiment is over the bullseye would have turned a whole circle. The distance travelled is the length of the 4 metre rulers plus the circumference of the circle. We had several more good solutions for the first part of this problem from pupils at Highcliffe Primary School. Whitney and Joe said: The new shape (i.e. the locus of the centre of the circle) will be a square with rounded corners. Sam and John explained that the length of the locus ... <div>... is the perimeter of the old square plus that of the circle.</div> <div>Oliver from Olchfa School sent us <a class="editorial" href="https://nrich.maths.org/content/solution/id/2159/W81khP-Solution%20to%20Roundabout.pdf"> this</a> clearly argued solution with useful diagrams which went beyond squares and generalised for circles rolling round any convex polygons.</div> <div class="editorial">Bill from Dana Middle School also gave us a detailed description of the path of the centre of the circle:</div> <div>As the circle rolls along the outside of the square, its centre traces a path one radius distant from the side of the box, parallel to the side, and the same length as the side, until the circle gets to a corner. At the corner, the point on the circle stays fixed to the corner, while the centre of the circle traces a circular path of radius r around the corner, until the radius (from the corner of the square, now) has swept out a certain angle (in this case, 90 degrees). The sweep begins with the radius at a right angle to the side the circle is leaving, and ends when the radius is at a right angle to the side it is entering. Since it does this four times, the length of the path around the corners of the square equals the circumference of the circle.</div> In effect, in circumnavigating the corners, the centre of the circle rotates completely around a point on the circle's circumference one time, and, since the distance from the centre to the edge is the same as the distance from the edge to the centre, this part of the path is the length of the circumference. On the straight sections, the length of the path is equal to the length of the sides, making it equal to the perimeter of the square. The length of the path is the perimeter of the square plus the circumference of the circle. By like reasoning, when the circle rolls around a triangle, as it rounds the corners the circle will turn around completely once, which results in the centre of the circle tracing out the circumference, which, when added to the perimeter of the triangle will give the length of the path. In general, as a circle rolls around a convex polygon, the length of the path of the centre of the circle will be the perimeter of the polygon plus the circumference of the circle, the locus of the path being a distance away from the polygon equal to the radius of the circle. Very thorough explanations; thank you Oliver and Bill . </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> How about asking pupils to imagine this "in their mind's eye" - no paper, no body language and lots of discussion? Alternatively this can be tackled through a practical activity involving a shape and a coin. It is not necessary to be able to calculate the circumference of a circle. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> What path does the centre of the circle take as it travels along one side of the shape? What shape will the path of the centre be as it goes round a corner? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had several great solutions for the first part of this problem from pupils at Highcliffe Primary School. Whitney and Joe said: <div style="margin-left: 40px;">The new shape (i.e. the locus of the centre of the circle) will be a square with rounded corners.</div> Sam and John explain that the length of the locus ...: <div style="margin-left: 40px;">... is the perimeter of the old square plus that of the circle.</div> Bill from Dana Middle School has given us a more detailed description of the path of the centre of the circle: <div style="margin-left: 40px;">As the circle rolls along the outside of the square, its centre traces a path one radius distant from the side of the box, parallel to the side, and the same length as the side, until the circle gets to a corner. At the corner, the point on the circle stays fixed to the corner, while the centre of the circle traces a circular path of radius r around the corner, until the radius (from the corner of the square, now) has swept out a certain angle (in this case, 90 degrees). The sweep begins with the radius at a right angle to the side the circle is leaving, and ends when the radius is at a right angle to the side it is entering. Since it does this four times, the length of the path around the corners of the square equals the circumference of the circle.</div> <div style="margin-left: 40px;">In effect, in circumnavigating the corners, the centre of the circle rotates completely around a point on the circle's circumference one time, and, since the distance from the centre to the edge is the same as the distance from the edge to the centre, this part of the path is the length of the circumference. On the straightaways, the length of the path is equal to the length of the sides, the total is the perimeter of the square.</div> <div style="margin-left: 40px;">The length of the path is the perimeter of the square plus the circumference of the circle.</div> <div style="margin-left: 40px;">By like reasoning, when the circle rolls around a triangle, as it rounds the corners the circle will turn around completely once, which results in the centre of the circle tracing out the circumference, which, when added to the perimeter of the triangle will give the length of the path.</div> <div style="margin-left: 40px;">In general, as a circle rolls around a convex polygon, the length of the path of the centre of the circle will be the perimeter of the polygon plus the circumference of the circle, the locus of the path being a distance away from the polygon equal to the radius of the circle.</div> A very thorough explanation, thank you Bill. </mdoxml> 2 4 0 0 1 0 0 Rolling Around A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle? Using, Applying and Reasoning about Mathematics Visualising Measures and Mensuration Length/distance Sequences, Functions and Graphs Loci 2D Geometry, Shape and Space Circles Information and Communications Technology Interactivities Secondary Mapping Document MD Construction and Loci Secondary Mapping Document DisplayCabinet Secondary processes PM - Visualising