Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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.
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Points A, B and C are the centres of three circles, each one of
which touches the other two. Prove that the perimeter of the
triangle ABC is equal to the diameter of the largest circle.
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?
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
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. . . .
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.
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred 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? ...
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. . . .
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
The three corners of a triangle are sitting on a circle. The angles
are called Angle A, Angle B and Angle C. The dot in the middle of
the circle shows the centre. The counter is measuring the size. . . .
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
Can you maximise the area available to a grazing goat?
What is the same and what is different about these circle
questions? What connections can you make?
How efficiently can you pack together disks?
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?
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
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
This article for pupils gives some examples of how circles have featured in people's lives for centuries.
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?
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?
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
What fractions of the largest circle are the two shaded regions?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Bluey-green, white and transparent squares with a few odd bits of
shapes around the perimeter. But, how many squares are there of
each type in the complete circle? Study the picture and make. . . .
See if you can anticipate successive 'generations' of the two
animals shown here.
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.
Introducing a geometrical instrument with 3 basic capabilities.
Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
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?
Given a square ABCD of sides 10 cm, and using the corners as
centres, construct four quadrants with radius 10 cm each inside the
square. The four arcs intersect at P, Q, R and S. Find the. . . .
A metal puzzle which led to some mathematical questions.
Construct this design using only compasses
Can Jo make a gym bag for her trainers from the piece of fabric she has?