Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Surface Areas and Volumes ↓


Volume of a Prism


Understanding prism

A prism is a three-dimensional figure consisting of two parallel faces that are identical, called bases, and rectangular sides connecting these bases. The size and shape of the base determine the name of the prism. If the base is a triangle, it is called a triangular prism. If the base is a rectangle, it is called a rectangular prism, and so on.

Prisms are very common in our everyday life. Consider a chocolate bar that is divided equally into rectangular pieces, or a long rectangular box, these are examples of rectangular prisms.

Example 1: Identifying prism shapes

Look at the following figures and identify what type of prisms they are:

a) A tent in the shape of a triangular prism.
b) A cereal box, which looks like a rectangular prism.
c) Toblerone chocolate, which is an example of a triangular prism.

Volume concept

The volume of a solid is the space it occupies. For example, the volume of water in a vessel is determined by the space it occupies inside. For a prism, the volume is determined by the area of its base and its height.

To better understand volume, imagine filling the shape with unit cubes (cubes with 1 unit side on each side). The total number of these unit cubes will represent the volume of the prism.

Volume formula for prism

The formula for finding the volume of a prism is:

Volume = Base Area × Height

Here, the "base area" is the area of the base of the prism, and the "height" is the perpendicular distance between the two bases.

Top Face Lower face Height

Calculating volume - step by step

Follow these steps to find the volume of a prism:

Step 1: Determine the base area

First, find the area of the base. The base can be of many shapes such as triangle, rectangle, etc. Use the appropriate formula for the base shape.

For a triangular base with base width b and height h_b :

    Base Area (Triangle) = 0.5 × b × h_b

For a rectangular base of length l and width w :

    Base Area (Rectangle) = l × w

Step 2: Measure the height of the prism

Find the height of the prism, which is the distance between two equal bases.

Step 3: Apply the volume formula

Calculate the volume by multiplying the base area by the height of the prism.

Example:

Consider a rectangular prism with a base measuring 4 units by 3 units and a height of 5 units.

Calculate the base area:

Base Area = 4 units × 3 units = 12 square units

Now, calculate the volume:

Volume = Base Area × Height = 12 square units × 5 units = 60 cubic units

More examples

Example 2: Triangular prism

Find the volume of a triangular prism whose triangular base is 6 cm long and 4 cm high, and the height of the prism is 10 cm.

Find the area of the triangular base:

    Base Area = 0.5 × 6 cm × 4 cm = 12 cm²

Calculate the volume:

    Volume = Base Area × Height = 12 cm² × 10 cm = 120 cm³

Example 3: Cylindrical prism

Find the volume of a cylindrical prism (commonly called a cylinder) of radius 3 m and height 7 m.

The base in this case is a circle, so we use the formula for the area of a circle:

    Base Area = π × r² = π × (3 m)² = 9π m²

Volume of a cylindrical prism:

    Volume = Base Area × Height = 9π m² × 7 m = 63π m³

Special cases: right vs. oblique prisms

A right-angled prism is one in which the sides are perpendicular to the base, that is, the sides are rectangular. In an oblique prism the bases are not aligned directly above each other, but the volume formula remains the same because the cross-sections parallel to the base remain constant with height.

Conclusion

Understanding the volume of a prism equips you with the ability to calculate the space it occupies, which is essential in a variety of fields, including architecture, engineering, and everyday tasks like packing boxes or filling containers. Using the formula Volume = Base Area × Height, you can find the volume of any prism, provided you know its base shape.

Practice finding volume with different base shapes and understand how volume changes when the base area or height changes. This exploration will solidify your learning and application of the prism volume concept.


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Volume of a Prism

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