A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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

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

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?

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

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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

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

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?

Can you minimise the amount of wood needed to build the roof of my garden shed?

Can you work out the area of this isosceles right angled triangle?

Two ribbons are laid over each other so that they cross. Can you find the area of the overlap?

A palm tree has snapped in a storm. What is the height of the piece that is still standing?

How much of the inside of this triangular prism can Clare paint using a cylindrical roller?

A circle of radius 1 is inscribed in a regular hexagon. What is the perimeter of the hexagon?

Can you find the distance from the well to the fourth corner, given the distance from the well to the first three corners?

When you pull a boat in using a rope, does the boat move more quickly, more slowly, or at the same speed as you?

How do these measurements enable you to find the height of this tower?