Volume and surface area (3)

Geometry Level pending

All the faces of a right pyramid are equilateral triangles. The sides of all the triangles are of the same length. The centers of all the faces are joined to form a three dimensional figure. The volume of this figure is 4. What is the volume of the pyramid?

The answer is 108.

1 solution

Chris Lewis
May 18, 2019

The original shape is a regular tetrahedron. The smaller shape created by joining the centres of its faces is also a regular tetrahedron.

There are various ways to establish the scale factor. One simple way is as follows: recall (or work out) that the centroid of a triangle divides each of its medians in the ratio $1:2$ . It's then easy to see that the height of the smaller tetrahedron is one third of the height of the original tetrahedron, and its volume one twenty-seventh. So the original tetrahedron has volume $27\cdot4=\boxed{108}$ .

Chris, may I ask you to check whether the following problem is correctly framed? The centers of all the faces of a four dimensional hypercube are joined to form a four dimensional figure. What is the ratio of the volumes of the two? I have just framed this problem.

A Former Brilliant Member - 2 years ago

Interesting! I think there is some difficulty with wording, though, especially with the idea of faces. You may want to say "centres of all the (square) faces" (assuming that's what you mean).

You could list the number of points, edges, faces and cells to make it really clear, but I don't think it's necessary.

Also you'll need to define the ratio you want - perhaps ask for the ratio of the 4-D volume of the original shape to the 4-D volume of the new shape, or introduce some notation to make it as clear as possible. (If you had asked for the ratio of the volumes of the two shapes in the tetrahedron question, both $27$ and $\frac{1}{27}$ would have been valid answers).

I hope that helps!

Chris Lewis - 2 years ago

Thanks. Let me post it.

A Former Brilliant Member - 2 years ago

Volume and surface area (3)

The answer is 108.

1 solution

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