The number of faces (F), edges (E) and vertices (V) of a polyhedron are 7, 15…
2024
The number of faces (F), edges (E) and vertices (V) of a polyhedron are 7, 15 and x, respectively. Then, the value of (2F + 3E – 4x) is :
- A.
20
- B.
19
- C.
18
- D.
17
Show answer & explanation
Correct answer: B
Euler's formula for any convex polyhedron relates its faces (F), vertices (V), and edges (E) by the identity F + V − E = 2 — the sum of faces and vertices always exceeds the number of edges by exactly 2.
Substitute the given faces and edges into Euler's formula: F + V − E = 2 becomes 7 + V − 15 = 2.
Solve for the vertex count: V = 2 − 7 + 15 = 10.
Substitute F = 7, E = 15, and V = 10 into the required expression 2F + 3E − 4V.
Evaluate term by term: 2(7) = 14, 3(15) = 45, and 4(10) = 40, so 2F + 3E − 4V = 14 + 45 − 40 = 19.
Cross-check: substitute V = 10 back into Euler's formula — F + V − E = 7 + 10 − 15 = 2, which matches the identity, confirming V = 10 and hence the value 19.