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

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?

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

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?

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.

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?

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

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?

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

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.

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

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

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

A collection of short problems on Pythagoras's Theorem and Trigonometry.