Loci Resources

Age 14 to 16 Challenge Level:

This resource contains a range of problems and interactivies on the theme of loci. Some of the resources, such as Roundabout and Rollin' Rollin' Rollin', enable you to change the settings and therefore open up lots of opportunity for further investigations and extensions to the problem posed in the text.

Turning triangles

In this problem you are asked to think about a triangle rolling along a horizontal line. Describe the paths of each of the vertices and the relationships between them and the original triangle.


Start by considering the locus of hte centre of a circle as it rolls around a square. What happens when the circle rolls around different polygons? How about different polygons of different sizes? This interactivity offers the flexibility to change the number os sides of he poygon and the sizes of the polygon and rolling circle.

Is there a theorem?

One square slides around another of the same size maintaining contact and keeping the same orientation. How far does a dot on the sliding square travel? Investigate this problem with different sized squares and then consider different polygons.

Rollin' rollin' rollin'

In this problem - two circles of equal radius kiss at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P? The interactivity also offers opportunity to extend the investigation to other polygons.

The next two problems are more challenging and will extend your understanding of loci .

The Line and Its Strange Pair

In this problem the points P and P' are connected by the following rule:
P' can move to different places along a dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along the straight line what does P do and can you explain what the rule that connects them is?

Mapping the Wandering Circle

In this problem the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do and can you explain what the rule that connects them is?