Problem 53 - Algebra 1: Concepts and Skills

Question

A fireproof safe is designed in the shape of a cube. The length of each edge of the cube is 2 meters. What is the volume of the fireproof safe?

Step-by-Step Solution

Answer

The volume of the cube is 8 cubic meters.

Step 1: Identify the length of the edge

The length of the edge of the cube given in the problem is 2 meters.

Step 2: Identify the formula for the volume of a cube

The volume \(V\) of a cube is calculated using the formula \(V=a^3\), where \(a\) is the length of one edge.

Step 3: Substitute the length of the edge into the volume formula

Substitute the length of the edge \(a=2\) meters into the formula. It becomes \(V=2^3\).

Step 4: Calculate the volume

The calculation of the volume becomes \(V=2^3=8\).

Key Concepts

Cube

A cube is a three-dimensional geometric shape that features six equal square faces. Each face is a perfect square, meaning all its sides are of equal length. Understanding cubes is essential because they are commonly found in everyday life, from dice to storage boxes.

When speaking of a cube, there are specific properties that define it:

• All edges of a cube are of equal length.
• Each angle between two adjoining faces is a right angle (90 degrees).
• Cubes fall under the category of polyhedra, which are three-dimensional shapes with flat faces.

The defining feature of a cube is the symmetry across its faces, edges, and vertices, contributing to its even and orderly appearance. This intrinsic symmetry plays a significant role in geometric problem-solving and calculations.

Volume Calculation

When calculating the volume of a cube, the process involves determining how much space the shape occupies in a three-dimensional context. Volume is measured in cubic units. For a cube, the formula to find its volume is straightforward:

\[ V = a^3 \]

where \( V \) is the volume and \( a \) is the length of one edge of the cube. This formula arises because a cube is defined geometrically by its equal lengths across all sides, multiplying these values together captures the entirety of the space the cube encloses. For example, if the edge length of a cube is 2 meters, plugging that number into the formula gives:

\[ V = 2^3 = 8 \]

This simple calculation tells us that the cube has a volume of 8 cubic meters.

Geometric Shapes

Geometric shapes are fundamental forms that are defined by their boundaries and dimensions. They can be two-dimensional like squares and circles, or three-dimensional like cubes and spheres. Understanding these shapes is a core part of geometry and aids in various computational techniques.

Three-dimensional shapes, like a cube, incorporate concepts of height, width, and depth, which makes them unique from their two-dimensional counterparts. Here are other common three-dimensional shapes:

• Spheres: Round in shape, with all points on the surface equidistant from the center.
• Cylinders: Consist of two parallel circular bases connected by a curved surface.
• Pyramids: Have a flat polygonal base and triangular faces that converge to a point.

These shapes each have their volume calculation formulas adapting to their specific dimensions and geometric properties, emphasizing their role in broader mathematical contexts.