What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

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.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Describe how to construct three circles which have areas in the ratio 1:2:3.

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?

Can you explain what is happening and account for the values being displayed?

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

How far should the roof overhang to shade windows from the mid-day sun?

Trigonometry, circles and triangles combine in this short challenge.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

The second of three articles on the History of Trigonometry.

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?

The third of three articles on the History of Trigonometry.

How can you represent the curvature of a cylinder on a flat piece of paper?

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China.

If you were to set the X weight to 2 what do you think the angle might be?

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

Prove Pythagoras' Theorem for right-angled spherical triangles.

There are many different methods to solve this geometrical problem - how many can you find?

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

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.

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

How would you design the tiering of seats in a stadium so that all spectators have a good view?

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?