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Blue and White
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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Arclets
Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".
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Marbles in a box
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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Hexy-Metry
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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Three by One
There are many different methods to solve this geometrical problem - how many can you find?
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Triangles and petals
An equilateral triangle rotates around regular polygons and
produces an outline like a flower. What are the perimeters of the
different flowers?
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Tilted Squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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On the Edge
If you move the tiles around, can you make squares with different coloured edges?
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Sending a Parcel
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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Where to Land
Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?
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Right angles
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
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Trapezium Four
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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Cola Can
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
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Can they be equal?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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Curvy areas
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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Which solids can we make?
Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?
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Vector journeys
Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
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Perimeter Possibilities
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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Square coordinates
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
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Circles in quadrilaterals
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
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Triangle midpoints
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
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Opposite vertices
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
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Semi-regular Tessellations
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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Cuboid Challenge
What's the largest volume of box you can make from a square of paper?