Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?
Small circles nestle under touching parent circles when they sit on
the axis at neighbouring points in a Farey sequence.
A cheap and simple toy with lots of mathematics. Can you interpret
the images that are produced? Can you predict the pattern that will
be produced using different wheels?
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
See if you can anticipate successive 'generations' of the two
animals shown here.
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
How efficiently can you pack together disks?
A metal puzzle which led to some mathematical questions.
This pattern of six circles contains three unit circles. Work out
the radii of the other three circles and the relationship between
This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF.
Similarly the largest. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
In LOGO circles can be described in terms of polygons with an
infinite (in this case large number) of sides - investigate this
Given any three non intersecting circles in the plane find another
circle or straight line which cuts all three circles orthogonally.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Introducing a geometrical instrument with 3 basic capabilities.
Three semi-circles have a common diameter, each touches the other
two and two lie inside the biggest one. What is the radius of the
circle that touches all three semi-circles?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
M is any point on the line AB. Squares of side length AM and MB are
constructed and their circumcircles intersect at P (and M). Prove
that the lines AD and BE produced pass through P.
Can Jo make a gym bag for her trainers from the piece of fabric she has?
A small circle fits between two touching circles so that all three
circles touch each other and have a common tangent? What is the
exact radius of the smallest circle?
Four circles all touch each other and a circumscribing circle. Find
the ratios of the radii and prove that joining 3 centres gives a
What is the sum of the angles of a triangle whose sides are
circular arcs on a flat surface? What if the triangle is on the
surface of a sphere?
Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
Thinking of circles as polygons with an infinite number of sides -
but how does this help us with our understanding of the
circumference of circle as pi x d? This challenge investigates. . . .
Can you reproduce the design comprising a series of concentric
circles? Test your understanding of the realtionship betwwn the
circumference and diameter of a circle.
Construct this design using only compasses
Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
By inscribing a circle in a square and then a square in a circle
find an approximation to pi. By using a hexagon, can you improve on
Where should runners start the 200m race so that they have all run the same distance by the finish?
The ten arcs forming the edges of the "holly leaf" are all arcs of
circles of radius 1 cm. Find the length of the perimeter of the
holly leaf and the area of its surface.
Triangle ABC has altitudes h1, h2 and h3. The radius of the
inscribed circle is r, while the radii of the escribed circles are
r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1
+. . . .
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
We have four rods of equal lengths hinged at their endpoints to
form a rhombus ABCD. Keeping AB fixed we allow CD to take all
possible positions in the plane. What is the locus (or path) of the
point. . . .
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
Equal circles can be arranged so that each circle touches four or
six others. What percentage of the plane is covered by circles in
each packing pattern? ...
What is the same and what is different about these circle
questions? What connections can you make?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
A circle touches the lines OA, OB and AB where OA and OB are
perpendicular. Show that the diameter of the circle is equal to the
perimeter of the triangle
Investigate constructible images which contain rational areas.
Draw three equal line segments in a unit circle to divide the
circle into four parts of equal area.