Pythagoras' Theorem and Trigonometry: Age 11-16

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

Can you make sense of these three proofs of Pythagoras' Theorem?

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 work out the dimensions of the three cubes?

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.

The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?

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. How far along the bank should she land?

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

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.