Question

In a rectangular parallelepiped with a square base and the altitude \(h\), a cross section through two opposite lateral edges is drawn. Compute the total surface area of the parallelepiped, if the area of the cross section equals \(S\). e.

Short answer

The total surface area is \(\frac{S^2}{h^2} + \frac{4Sh}{\sqrt{2}}\).

Step-by-step solution

Understanding the Parallelepiped

A rectangular parallelepiped is a 3D rectangle, also known as a rectangular box. Here, we have a special case where the base is a square. Let's denote the side of the square base as \(a\) and the height of the parallelepiped as \(h\). The base is square, so both the length and width are \(a\).

Properties of the Cross Section

The cross section that passes through two opposite lateral edges will form a rectangle comprising the height \(h\) and the diagonal of the square base. Known as \(d\), the diagonal can be calculated using the formula for the diagonal of a square: \(d = a\sqrt{2}\).

Area of the Cross Section

We know that the area of the cross section is \(S\). The cross section is a rectangle with height \(h\) and base \(d = a\sqrt{2}\). This gives the area formula: \(h \, a\sqrt{2} = S\). Thus, we have: \(h a\sqrt{2} = S\).

Solve for the Side of the Base

From the equation \(ha\sqrt{2} = S\), solve for \(a\): \(a = \frac{S}{h\sqrt{2}}\). This is the side of the square base.

Compute the Total Surface Area

The surface area of a parallelepiped with a square base is composed of two square bases and four lateral rectangles. Each base has an area \(a^2\), and each lateral face has an area \(ah\). Therefore, Total Surface Area \(= 2a^2 + 4ah\).

Substitute to Find Surface Area

Substitute \(a = \frac{S}{h\sqrt{2}}\) into the total surface area expression: \[\begin{align*}&2\left(\frac{S}{h\sqrt{2}}\right)^2 + 4\left(\frac{S}{h\sqrt{2}}\right)h = \&2\frac{S^2}{h^2 \cdot 2} + 4\frac{S}{\sqrt{2}} = \&\frac{S^2}{h^2} + \frac{4Sh}{\sqrt{2}}.\end{align*}\]

Key concepts

Surface Area Calculation

Calculating the Surface Area of a rectangular parallelepiped is essential to understanding its total exterior coverage. A rectangular parallelepiped, sometimes simply called a box, has six faces. When it has a square base, the problem simplifies somewhat, as the rectangular nature of its faces can be handled easily with formulas.

The surface area of a rectangular parallelepiped is calculated by first considering each of its sides: two squares and four rectangles. The squares are from the base and the top, and they have equal area, calculated as the square of the side of the base, or \(a^2\). The rectangles, being the sides connecting the top and bottom bases, have an area defined as the base multiplied by the height, which means each is \(ah\), giving us four rectangles overall. Therefore, the total surface area \(A\) is \(2a^2 + 4ah\).

This simplicity comes from the symmetrical properties of the shape—it’s merely the sum of areas of its components.

Square Base

In the context of parallelepipeds, having a square base helps significantly simplify calculations. Imagine a 3-dimensional shape sitting flat with a perfect square for its bottom. Such a base means both length and width are equal; denoted generally as \(a\).

The importance of a square base lies in the consistency it brings to the shape. Both the bottom and the top faces are identical, making it easier to calculate things like the surface area and other geometrical properties. Because both faces are squares, their area is calculated as \(a^2\).

Understanding a square base is crucial when tackling related geometrical problems or variations, as it defines the initial parameters used in other formulations and calculations.

Cross Section Area

A cross section refers to the intersection of a plane with a solid, often used to simplify three-dimensional problems. For a rectangular parallelepiped with a square base, this cross section can be particularly insightful.

The task is often to understand the two main parameters defining this area: one side along the height \(h\) and another being the diagonal of the square base. The diagonal \(d\) can be found through the Pythagorean theorem for a square, calculated as \(a\sqrt{2}\). The cross-section forms a rectangular area with \(h\) and \(d\), meaning its area, \(S\), is \(ha\sqrt{2}\).

Determining the cross-sectional area is critical not only for understanding the inherent properties of the parallelepiped but also for indirect calculations such as finding the base side when this area is known.