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 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
What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?
The Earth is further from the Sun than Venus, but how much further?
Twice as far? Ten times?
How can you represent the curvature of a cylinder on a flat piece of paper?
What is an AC voltage? How much power does an AC power source
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.
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?. . . .
Solve the equation sin z = 2 for complex z. You only need the
formula you are given for sin z in terms of the exponential
function, and to solve a quadratic equation and use the logarithmic
Can you explain what is happening and account for the values being
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 ?
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.
There are many different methods to solve this geometrical problem - how many can you find?
Two problems about infinite processes where smaller and smaller
steps are taken and you have to discover what happens in the limit.
Join some regular octahedra, face touching face and one vertex of
each meeting at a point. How many octahedra can you fit around this
An environment that simulates a protractor carrying a right- angled
triangle of unit hypotenuse.
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.
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. . . .
Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?
In this 'mesh' of sine graphs, one of the graphs is the graph of
the sine function. Find the equations of the other graphs to
reproduce the pattern.
Find the exact values of some trig. ratios from this rectangle in
which a cyclic quadrilateral cuts off four right angled triangles.
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?