Question

Consider 4 spheres of radius 10 A. If they are used to form a regular tetrahedron, then calculate the height of
tetrahedron formed by joining their centers​

Answer 1

Given : Consider 4 spheres of radius 10 A. If they are used to form a regular tetrahedron.

To find : The height of tetrahedron formed by joining their centers.

solution : The base of tetrahedron is an equilateral triangle of side length 2r.

where r is the radius of sphere.

now distance from a vertex to centroid of triangle, x = 2r/√3 [ see figure ].....(1)

now The height of tetrahedron find from Pythagoras theorem.

(2r)² = x² + H²

⇒4r² = (2r/√3)² + H²

⇒H² = 4r² - 4r²/3

⇒H² = 8r²/3

⇒H = 2√(2/3)r

now putting r = 10 A°

so, the height of tetrahedron, H = 2√(2/3) × 10

= 20 × √(2/3) = 8.165 A°

Therefore The height of tetrahedron is 8.165 A°