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?
This article for pupils gives some examples of how circles have featured in people's lives for centuries.
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
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
Can you maximise the area available to a grazing goat?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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?
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 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?
Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.
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. . . .
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?
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.
Where should runners start the 200m race so that they have all run the same distance by the finish?
Investigate constructible images which contain rational areas.
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
Can Jo make a gym bag for her trainers from the piece of fabric she has?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What is the same and what is different about these circle
questions? What connections can you make?
Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?
See if you can anticipate successive 'generations' of the two
animals shown here.
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.
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. . . .
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
In LOGO circles can be described in terms of polygons with an
infinite (in this case large number) of sides - investigate this
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?
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?
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?
What fractions of the largest circle are the two shaded regions?
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. . . .
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.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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. . . .
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
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 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?
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. . . .
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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. . . .
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).
Construct this design using only compasses
A metal puzzle which led to some mathematical questions.
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.
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?
The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .
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. . . .
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?