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Holly and Carol make Christmas decorations by sticking a square based pyramids to each square in the net of a cube. The net is made from 6 cm squares and the pyramids are 3 cm high so that when the net is folded up the pyramids fill the cube. Holly and Carol turn the net inside out so that the pyramids are on the outside of the cube. Show that the shape has 12 faces and calculate its volume. Make a decoration of your own – you will need to calculate the measurements of the triangles that make the nets of the pyramids.

Each pyramid has 4 triangular faces,

so the decoration seems to have \(6 \times 4 = 24\) faces.

In this cross section of the decoration B and D midpoints of edges of cube

C is the midpoint of the square face.

AB = BC = 3 cm angle ACD = 90°

So angle ABC = 45°

Similarly angle EBG = 45°

So ABE is a straight line.

So the two triangles that share an edge are one face.

So the decoration has \(24 \div 2 = 12\) faces.

The total volume of the 6 pyramids is the same as the cube

So the total for the decoration is \(2 \times 6^{3} = 432\) cm\(^3\)

To find the shape of the triangular face of each pyramid

From triangle ABC, the length AB is \(\sqrt{3^{2} + 3^{2}} = 3\sqrt{2}\) cm

XY is the edge of a square so XB = 3 cm

So AX = AY = \(\sqrt{(3\sqrt{2})^{2} + 3^{2}} = 3\sqrt{3} \approx 5.2\) cm

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