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