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?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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.

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. . . .

Can you make sense of these three proofs of Pythagoras' Theorem?

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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. . . .

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Can you find the areas of the trapezia in this sequence?

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

This is the second article on right-angled triangles whose edge lengths are whole numbers.

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

What fractions can you divide the diagonal of a square into by simple folding?

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

A huge wheel is rolling past your window. What do you see?

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

Here are some examples of 'cons', and see if you can figure out where the trick is.

If you think that mathematical proof is really clearcut and universal then you should read this article.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

Four jewellers share their stock. Can you work out the relative values of their gems?

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Keep constructing triangles in the incircle of the previous triangle. What happens?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Prove Pythagoras' Theorem using enlargements and scale factors.

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.