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?
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?
What fractions of the largest circle are the two shaded regions?
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).
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. . . .
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. . . .
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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.
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.
This article for pupils gives some examples of how circles have featured in people's lives for centuries.
Can you maximise the area available to a grazing goat?
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. . . .
Introducing a geometrical instrument with 3 basic capabilities.
This interactivity allows you to sort logic blocks by dragging their images.
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?
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 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 metal puzzle which led to some mathematical questions.
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 Yin Yang symbol using a pair of compasses?
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?
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. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
What shape is the overlap when you slide one of these shapes half
way across another? Can you picture it in your head? Use the
interactivity to check your visualisation.
If these balls are put on a line with each ball touching the one in
front and the one behind, which arrangement makes the shortest line
Can Jo make a gym bag for her trainers from the piece of fabric she has?
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
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?
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.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
In LOGO circles can be described in terms of polygons with an
infinite (in this case large number) of sides - investigate this
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.
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?