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Qqq..cubed

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

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In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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Mouhefanggai

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Concrete Calculation

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. The shape is designed to avoid subsidence that might be caused by the roots of some very large trees nearby. The hole is perfectly symmetrical with vertical walls, 1.2 metres high, surrounding a rectangular area measuring 6 metres by 5 metres. A mound of earth in the centre, the shape of the frustum of a pyramid, has a rectangular base 4.8 metres by 3.8 metres and a horizontal rectangular top 3.8 metres by 2.8 metres. The height of this mound is 0.9 metres. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations? Can you explain the method simply to help the builders to do the calculations for similar shaped rafts of other dimensions?