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Let ABCD  be a regular tetrahedron. Let E,F , G,H  be the centers of faces BCD , ACD,ABD,ABC respectively. The volume of pyramid ABCD is 18 Find the volume of pyramid EFGH .

 May 4, 2024
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Let's denote the side length of the tetrahedron as a.

 

Volume of Tetrahedron ABCD: The formula for the volume of a regular tetrahedron is V = (a^3 * √2) / (12). We are given that V = 18, so:

 

(a^3 * √2) / (12) = 18 a^3 * √2 = 216 a^3 = 216 / √2 (rationalize the denominator by multiplying top and bottom by √2) a^3 = 216√2 / 2 a^3 = 108√2

 

Volume of Pyramid EFGH: EFGH is a smaller tetrahedron similar to ABCD. The ratio of side lengths between similar figures is the same as the ratio of their corresponding edge lengths.

 

In this case, the edges of EFGH are half the length of the edges of ABCD because E, F, G, and H are the centers of the faces. Therefore, the side length of EFGH is a/2.

 

We can now find the volume of EFGH using the same formula for a tetrahedron, but with the side length a/2:

 

V_EFGH = [(a/2)^3 * √2] / (12) V_EFGH = (a^3 * √2) / (48)

 

Relating the Volumes: We know a^3 = 108√2 from part 1. Substitute this into the formula for V_EFGH:

 

V_EFGH = [(108√2) * √2] / (48)

 

V_EFGH = (108 * (√2)^2) / (48)

 

V_EFGH = (108 * 2) / (48)

 

V_EFGH = 216 / 48 = 9/2

 

Therefore, the volume of pyramid EFGH is 9/2.

 May 4, 2024

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