Sine rule and cosine rule

problem
Calculating with cosines

Age

14 to 18

Challenge level

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?
problem
Raising the roof

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
problem
Cyclic triangles

Age

16 to 18

Challenge level

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.
problem
Xtra

Age

14 to 18

Challenge level

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.
problem
Three by one

Age

16 to 18

Challenge level

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

Age

14 to 16

Challenge level

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
problem
Pythagoras for a tetrahedron

Age

16 to 18

Challenge level

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.
problem
Darts and kites

Age

14 to 16

Challenge level

Explore the geometry of these dart and kite shapes!
problem
30-60-90 polypuzzle

Age

16 to 18

Challenge level

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