Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Mensuration → Surface Area and Volume ↓


Cubes and Cuboids


Understanding the concepts of cube and cuboid is important to learn about shapes and space in measurement. These are three-dimensional objects that have length, width, and height. Mathematics is the branch of mathematics that deals with the measurement of geometric shapes and figures, such as length, area, and volume. In this article, we will discuss the nuances of cube and cuboid in depth and find out how to calculate their surface area and volume.

Introduction to cubes and cuboids

First, let's figure out what a cube and a cuboid are:

  • Cuboid: A cuboid is a three-dimensional shape with six rectangular faces. Each of its angles is a right angle. It looks like a box or rectangular container.
  • Cube: A cube is a special type of cuboid in which all three dimensions (length, breadth and height) are equal. This makes all the faces of the cube square.

Visual representation of cuboid

The above figure is a visual representation of a cuboid.

Visual representation of a cube

The above picture is a visual representation of a cube.

Surface area of cuboid

A cuboid has six faces, with opposite faces being identical. The total surface area of a cuboid is the sum of the areas of all its six rectangular faces. If a cuboid has a length l, width b and height h, its surface area is calculated as:

Total Surface Area = 2(lb + bh + hl)

For example, let us find the surface area of a cuboid with length 5 cm, width 3 cm, and height 4 cm.

Total Surface Area = 2(5 * 3 + 3 * 4 + 4 * 5)
Total Surface Area = 2(15 + 12 + 20)
Total Surface Area = 2(47)
Total Surface Area = 94 square cm

Surface area of a cube

Since a cube is essentially a cuboid with all sides equal, each face of the cube is a square. If the side of the cube is a, then the surface area of the cube is:

Total Surface Area = 6a2

For example, if the side of a cube is 4 cm, then its surface area will be:

Total Surface Area = 6 × 42
Total Surface Area = 6 × 16
Total Surface Area = 96 square cm

Volume of a cuboid

The volume of a cuboid is the space it occupies, it can be determined by multiplying its length, width and height. If a cuboid has a length l, width b and height h, then its volume is:

Volume = l × b × h

For example, let's find the volume of a cuboid with a length of 5 cm, a width of 3 cm, and a height of 4 cm:

Volume = 5 × 3 × 4
Volume = 60 cubic cm

Volume of a cube

Since a cube is a special type of cuboid where all sides are equal, the volume of a cube is obtained by multiplying the length of the side by itself three times. If the side of a cube is a, then its volume is given by:

Volume = a3

For example, if the side length of a cube is 4 cm, then its volume will be:

Volume = 43
Volume = 64 cubic cm

Example problems

Example problem 1: Surface area of cuboid

Find the total surface area of a cuboid whose length is 8 cm, width is 6 cm and height is 5 cm.

Use of the formula:

Total Surface Area = 2(lb + bh + hl)

Substitute Values:

Total Surface Area = 2(8 * 6 + 6 * 5 + 5 * 8)
Total Surface Area = 2(48 + 30 + 40)
Total Surface Area = 2(118)
Total Surface Area = 236 square cm

Example problem 2: Volume of a cube

Find the volume of a cube of side 7 cm.

Use of the formula:

Volume = a3

Substitute Values:

Volume = 73
Volume = 343 cubic cm

Example problem 3: Volume of a cuboid

Find the volume of a cuboid of length 3 cm, breadth 4 cm and height 5 cm.

Use of the formula:

Volume = l × b × h

Substitute Values:

Volume = 3 × 4 × 5
Volume = 60 cubic cm

Example problem 4: Surface area of a cube

Find the surface area of a cube of side 6 cm.

Use of the formula:

Total Surface Area = 6a2

Substitute Values:

Total Surface Area = 6 × 62
Total Surface Area = 6 × 36
Total Surface Area = 216 square cm

Important things to remember

  • A cube is a special form of a cuboid, where the length, breadth and height are equal.
  • The total surface area of a cube can be found using the formula 6a2.
  • The surface area of a cuboid is found by adding the areas of its six faces.
  • Volume measures the space occupied by an object; for cubes and cuboids, it is expressed in cubic units.
  • The volume and surface area formulas are important for solving measurement problems.

Conclusion

Understanding cubes and cuboids helps deal with problems related to space and capacity. By knowing the surface area and volume formulas, one can tackle many practical problems. Whether you are calculating how much material is needed to build a box or how much space is left in a storage container, these concepts are incredibly useful.


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Cubes and Cuboids

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