Finding Cuboid Volume from Adjacent Face Areas

The question asks us to find the volume of a cuboid given the surface areas of three adjacent faces. Let's consider a cuboid with dimensions length (l), width (w), and height (h).

A cuboid has six rectangular faces. Adjacent faces share an edge. The three adjacent faces whose areas are given would share a common vertex. Their areas can be represented as the products of pairs of the cuboid's dimensions:

• Area of one face (e.g., front): length × height = lh
• Area of an adjacent face (e.g., side): width × height = wh
• Area of the third adjacent face (e.g., top): length × width = lw

Let the given surface areas of the three adjacent faces be a, b, and c. We can assign them as follows:

Let:

\(a = lw\)

\(b = wh\)

\(c = hl\)

We want to find the volume (V) of the cuboid. The formula for the volume of a cuboid is:

\(V = lwh\)

We need to express V in terms of a, b, and c. Let's multiply the three equations for a, b, and c together:

\(a \times b \times c = (lw) \times (wh) \times (hl)\)

\(abc = l \times w \times w \times h \times h \times l\)

\(abc = l^2 w^2 h^2\)

We can rewrite the right side of the equation:

\(abc = (lwh)^2\)

We know that the volume V is \(lwh\). So, we can substitute V into the equation:

\(abc = V^2\)

To find the volume V, we need to take the square root of both sides of the equation:

\(\sqrt{abc} = \sqrt{V^2}\)

\(V = \sqrt{abc}\)

This can also be written using fractional exponents:

\(V = (abc)^\frac{1}{2}\)

Thus, the volume of the cuboid is the square root of the product of the areas of the three adjacent faces.

Let's verify with a simple example. Consider a cuboid with l=2, w=3, h=4.

• Area of face lw = 2 × 3 = 6 (let a=6)
• Area of face wh = 3 × 4 = 12 (let b=12)
• Area of face hl = 4 × 2 = 8 (let c=8)

Product of adjacent areas \(abc = 6 \times 12 \times 8 = 576\)

The volume \(V = lwh = 2 \times 3 \times 4 = 24\)

According to our derived formula, \(V = \sqrt{abc} = \sqrt{576}\). Since \(24 \times 24 = 576\), \(\sqrt{576} = 24\). This matches the volume calculated directly from the dimensions.

Revision Table: Cuboid Formulas

Here's a quick look at key formulas for a cuboid with dimensions l, w, h:

Additional Information: Understanding Cuboids

A cuboid is a 3D shape with six rectangular faces, twelve edges, and eight vertices. All angles are right angles. It's also known as a rectangular prism or a right rectangular prism. If all edges are equal in length, the cuboid is a cube.

Understanding the relationship between the dimensions, surface areas, and volume of a cuboid is fundamental in geometry and mensuration. Adjacent faces provide insight into the pairwise products of dimensions, while the volume is the product of all three dimensions.

The method used above, multiplying the equations involving the dimensions, is a common technique to solve problems where products of variables are given and the product of all variables is required. In this case, squaring the desired quantity (volume) made it easier to relate it back to the given quantities (areas of adjacent faces).