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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?

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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.

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

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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.

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

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

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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?

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

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

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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?

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Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?

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

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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?

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