Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
Age 11 to 14 Challenge Level:
Well done toeveryone who correctly worked out that the
third solution in the hint is correct. From 20 feet above the
ground the bottom of the wheel appears horizontal, so it covers the
top right corner almost immediately after covering the top left
corner. This can be seen by using the interactivity and making the
wheel very large by pressing the up arrow:
Thanks to Julian (Hastingsbury School) who
worked out how long the window would be covered for.
If you consider the radius that touches the ground, the angle this
makes with the radius 20 feet above the ground is 0.7 degrees, so
the wheel turns 1.4 degrees whilst covering the window.
The wheeil travels 100pi miles each time it completes a whole turn,
taking pi hours. So the wheel will cover the window for (1.4 / 360)
pi hours = 0.0122 hours = 44 seconds. The angle of the tangent to
the circle where it first touches the window is less than 1 degree
to the horizontal, and so it appears to be coming down horizontally
as we expected above!
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.