Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Mensuration → Surface Area and Volume ↓


Cubes


In mathematics, understanding the concept of a cube is important when studying the surface area and volume of three-dimensional shapes. A cube is a special type of rectangular prism where all sides are the same length. This unique feature gives the cube its distinctive properties, making it an important shape in geometry and measurement.

What is a cube?

A cube is a three-dimensional solid object bounded by six square faces, facets, or sides, three of which meet at each vertex. The edges and corners of a cube are equal, making it a very regular and symmetrical shape. In other words, all edges have the same length. If the length of one edge of a cube is represented by s, then each face is a square with a side length of s.

Characteristics of the cube

  • All the faces of a cube are squares.
  • All edges have the same length.
  • A cube has 12 edges, 6 faces, and 8 vertices.
  • All angles in a cube are right angles (90 degrees).

Surface area of a cube

The surface area of a cube is the total area of all its six square faces. Since each face is a square and all squares have the same side length s, the area of one face is simply s^2. Since a cube has six equal faces, the formula for the total surface area A can be expressed as:

a = 6s^2

This formula helps us determine the amount of material needed to cover the entire surface of a cube.

Example:

Let's consider a cube with a side length of 4 cm. To find its surface area, we can use the surface area formula:

        S = 4 cm
        a = 6s^2
          = 6(4 cm)^2
          = 6(16 cm^2)
          = 96 cm^2

Thus, the surface area of this cube is 96 square centimeters.

Visualizing a cube

A cube can be easily visualized as a box where all sides are of equal length. Below is a visual depiction of a cube. Consider a cube whose side length is labeled as s.

S S S

Volume of a cube

The volume of a solid shape such as a cube is the space it occupies. To find the volume V of a cube, you need to multiply the length, width, and height, which are equal in a cube. Therefore, the formula for the volume of a cube is:

v = s^3

This formula tells us how much space there is inside the cube.

Example:

Consider a cube with a side length of 3 meters. To find its volume:

        S = 3 m
        v = s^3
          = (3 m)^3
          = 27 m^3

Thus, the volume of the cube is 27 cubic meters.

Applications of cubes

Understanding cubes is useful in many real-world contexts. Here are some applications:

  • Packaging: Cubes are commonly used in packaging because they maximize space-efficiency for storage and shipping.
  • Construction: The properties of the cube are often applied in architecture and construction, with buildings often containing cube-shaped rooms.
  • Education: Children learn about shapes using cubes because of their simplicity and uniformity.

Cube mesh

An important concept related to three-dimensional shapes is the net. The net of a cube is a two-dimensional shape that can be folded to form a cube. For a cube, the net consists of six squares connected in a T-shape, cross, or other valid configuration.

Here's a typical mesh for a cube:

This net can be given the shape of a cube by folding it at the edges.

Practice problems

Here are some practice problems to help reinforce the concepts of surface area and volume of cubes:

Problem 1:

Find the surface area of a cube whose side length is 7 meters.

Solution:

        S = 7 m
        a = 6s^2
          = 6(7 m)^2
          = 6(49 m^2)
          = 294 m^2

Thus, the surface area is 294 square meters.

Problem 2:

The volume of a cube is 64 cm3. Determine the length of its side.

Solution:

        v = s^3
        64 cm^3 = s^3
        S = ∛(64 cm^3)
        S = 4 cm

The length of the side of the cube is 4 cm.

Problem 3:

Find the volume of a cube of side 9 cm.

Solution:

        S = 9 cm
        v = s^3
          = (9 cm)^3
          = 729 cm^3

The volume of the cube is 729 cubic centimeters.


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Cubes

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