### Three Cubes

Can you work out the dimensions of the three cubes?

### In a Spin

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

### 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:

Several solutions arrived this month but some of you fell into the trap of assuming that the sides of the frustrum would converge to a point but they do not so the solution needed you do dissect the mound in the middle as suggested here by Michael Levitin of St. Margaret's Episcopal School San Juan Capistrano, California, USA. I like your advice to the builder Michael, how about a spreadhsheet?

The answer is 23.154 cubic meters.

The solution:

First we figure out the volume of the whole rectangular prism. Then we figure out the volume of the frustrum of the pyramid. Then we simply subtract the volume of the frustrum from the volume of the rectangular prism.

The volume of the rectangular prism is the area of the base times the height. So that is 6 x 5 x 12 = 36 cubic meters.

To calculate volume of the frustrum:

Put the frustum on a cutting board with the bigger base down. Take a huge sharp knife and make four straight cuts along the edges of the top square. Then we will have nine pieces.

And they will be:

One rectangular prism with base sides of 3.8 m by 2.8 m and with height of 0.9 m. Then the volume will be 3.8 x 2.8 x 0.9 = 9.576 cubic meters.

We will also have four triangular prisms from the sides of frustrum that we will put one on top of the other to form a triangular prism with the right triangle base with legs of 0.5 m and 0.9 m and area of (0.5 x 0.9)/2 = 0.225 square meters. The height of the triangular prism is the perimeter of the frustrum top base and is 2 x (3.8 + 2.8) = 13.2 m. So the volume of the triangular prism is the area of the base times height and equal to 0.225 x 13.2 = 2.97 cubic meters.

We will combine four corners of the frustrum and make a pyramid with a square base with sides of 1 m and a height of 0.9 m. Then the area of the base is 1 square meter and the volume is 1 x 0.9 / 3 = 0.3 cubic meters.

Then we find the volume of the frustrum as the sum of the volume of the figures. 9.576 + 2.97 + 0.3 = 12.846 cubic meters. Then we subtract the volume of the frustrum from the volume of the rectangular prism 36 - 12.846 = 23.154 cubic meters.

Suggestion to the builders: Either do what we have done or make a computer program that does it.