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?
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. . . .
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
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. . . .
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.
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.
This pattern of six circles contains three unit circles. Work out
the radii of the other three circles and the relationship between
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?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
A metal puzzle which led to some mathematical questions.
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.
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?
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? ...
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
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Given any three non intersecting circles in the plane find another
circle or straight line which cuts all three circles orthogonally.
Four circles all touch each other and a circumscribing circle. Find
the ratios of the radii and prove that joining 3 centres gives a
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?
Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?
Investigate constructible images which contain rational areas.
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
Small circles nestle under touching parent circles when they sit on
the axis at neighbouring points in a Farey sequence.
What is the same and what is different about these circle
questions? What connections can you make?
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.
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?
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?
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.
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
+. . . .
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. . . .
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 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?
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.
See if you can anticipate successive 'generations' of the two
animals shown here.
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?
Construct this design using only compasses
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.
Where should runners start the 200m race so that they have all run the same distance by the finish?
A security camera, taking pictures each half a second, films a
cyclist going by. In the film, the cyclist appears to go forward
while the wheels appear to go backwards. Why?
How efficiently can you pack together disks?
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.
In LOGO circles can be described in terms of polygons with an
infinite (in this case large number) of sides - investigate this
Draw three equal line segments in a unit circle to divide the
circle into four parts of equal area.
A spiropath is a sequence of connected line segments end to end
taking different directions. The same spiropath is iterated. When
does it cycle and when does it go on indefinitely?
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. . . .