Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?
Rotating a pencil twice about two different points gives surprising results...
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?
What shaped overlaps can you make with two circles which are the
same size? What shapes are 'left over'? What shapes can you make
when the circles are different sizes?
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.
Can you maximise the area available to a grazing goat?
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?
Two circles of equal radius touch 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 area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radii.
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.
This pattern of six circles contains three unit circles. Work out
the radii of the other three circles and the relationship between
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?
Six circles around a central circle make a flower. Watch the flower
as you change the radii in this circle packing. Prove that with the
given ratios of the radii the petals touch and fit perfectly.
Look at the different ways that Erika,
Phoebe, Alice and Luke, and Jas found to solve this problem.
It is interesting to compare the variety of approaches that students have used to solve it.
The three solutions from Rachael and Katy,
James and Derek show an interesting range of approaches and detail.
This solution treats the problem as open ended and gives an account of further work involving analagous situations and investigation of the topic using the web.
This article teaches you how to draw cardiods, limacons, nephroids and ellipses - a lot easier than they sound! All you need is a pair of compasses and a pencil.
Exchange the positions of the two sets of counters in the least possible number of moves
Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.
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?
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.