Question

If areas of three adjacent faces of a cuboid be x, y and z respectively, then find the volume of the cuboid.

Answer 1

Hint:
1) A cuboid is a three-dimensional solid whose all faces are rectangles.
2) There are six rectangular faces in a cuboid and every pair of opposite faces is identical.
3) The area of a rectangle is equal to $ L\times B $ sq. units, where L units is the length and B units is the breadth.
4) The volume of a cuboid is $ L\times B\times H $ cubic units.

Complete step by step solution:
Let's say that the length, breadth and the height of the cuboid are L, B and H respectively.
Since the adjacent faces are all unequal, their areas, according to the question, will be:
  $ L\times B=x $ ... (1)
  $ B\times H=y $ ... (2)
  $ H\times L=z $ ... (3)
Multiplying together the above equations, we get:
  $ {{\left( L\times B\times H \right)}^{2}}=xyz $
⇒ $ L\times B\times H=\sqrt{xyz} $

Hence, the volume of the cube is $ \sqrt{xyz} $ cubic units.

Note:
1) There are two main types of solids:
Polyhedron: It must have flat faces. e.g. Cube/Cuboid, Tetrahedron etc.
Non-Polyhedron: Some face(s) may be curved. e.g. Cone, Cylinder, Sphere etc.
2) Euler's Formula: For a polyhedron: $ F+V-E=2 $ , where F is the number of faces, V is the number of vertices and E is the number of edges.
3) There are 8 vertices, 12 edges and 6 faces in a cube/cuboid.