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Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Of All the Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
You may find it helpful to use the virtual geoboard environment for this problem.

When working on an isometric grid, we can measure areas in terms of equilateral triangles instead of squares.

Here are some equilateral triangles.

If the area of the first triangle is 1, what are the areas of the other triangles?

Can you see a relationship between the area and the length of the base of each triangle?

Will the pattern continue?
Can you explain why?

All the triangles in the first image had horizontal bases, but it is also possible to draw tilted equilateral triangles.

Here are some equilateral triangles with a tilt of 1.

Can you convince yourself that they are equilateral?

Can you find their areas?

Take a look at the hint for ideas on how to get started.

Can you find a rule to work out the area of any equilateral triangle with a tilt of 1?

Can you explain why your rule works?

What about triangles with a tilt of 2?

What about other tilts?