Volume of Prisms and Cylinders

In math, when we want to measure the volume of a solid figure, we first need to understand what the figure looks like and its structure. In grade 7 math, two important types of 3-dimensional shapes that we often come across are prisms and cylinders. These shapes are quite common and can be found in many everyday objects around us.

What is volume?

Volume is a measure of the space occupied by an object. It is usually measured in cubic units such as cubic centimeters (^cm3), cubic meters (^m3), etc. If you think of a box filled with cubes, the volume would be the count of all the small cubes present inside the box.

What are prisms?

A prism is a solid object with two equal ends and flat sides. The important thing to remember is that the cross-section does not change along its length. The ends or bases decide the name of the prism. For example, if the base is a triangle, it is called a triangular prism.

Examples of prisms

• triangular prism
• rectangular prism (also called cuboid)
• hexagonal prism

Visual example of a prism

Consider the following simplified visual representation of a rectangular prism:

The highlighted portion represents the top of the prism, which represents one of the bases. The other flat surfaces connect the two ends. The cross-section between the two ends remains the same throughout the shape.

How to calculate the volume of a prism?

The volume of a prism can be found by multiplying the area of the base of the prism by its height. In mathematical terms:

    Volume of prism = Base Area × Height

Example calculation of a triangular prism

Let's find the volume of a triangular prism. Suppose we have a prism whose base is a right-angled triangle with a length of 3 cm and a height of 4 cm. The height of the prism is 10 cm.

First, find the area of the triangular base:

    Area of base = 1/2 × Base of triangle × Height of triangle = 1/2 × 3 cm × 4 cm = 6 cm²

Now find the volume of the prism:

    Volume = Base Area × Height of prism = 6 cm² × 10 cm = 60 cm³

What is a cylinder?

A cylinder is a solid object with two equal circular bases connected by a curved surface. It also has a uniform cross-section along its length that is similar to a prism but with curved edges.

Visual example of a cylinder

The circular parts form the bases of the cylinder. The curved surface connects these two bases and wraps around them.

How to calculate the volume of a cylinder?

The volume of a cylinder is found by multiplying the area of the circular base of the cylinder by the height of the cylinder. The formula is:

    Volume of cylinder = π × Radius² × Height

Here, π (pi) is approximately 3.14159.

Example of calculation of cylinder

Let us find the volume of a cylinder with a circular base of radius 5 cm and height 12 cm.

First, find the area of the base:

    Area of base = π × Radius² = π × (5 cm)² = π × 25 cm² ≈ 78.54 cm²

Now find the volume of the cylinder:

    Volume = Area of Base × Height = 78.54 cm² × 12 cm ≈ 942.48 cm³

Comparison and conclusion

Although prisms and cylinders differ in terms of their base and lateral structures, the volume of both can be found by multiplying the base area by the height. Understanding the volume of these shapes helps us apply these concepts to real-world problems, whether it is figuring out how much stuff a container can hold or calculating the amount of material needed for construction.

A solid understanding of these concepts paves the way for more complex geometrical and mathematical studies as students progress in their education. Always remember, practicing with different examples can greatly enhance understanding and master these essential mathematical concepts.

Keep experimenting with different shapes of bases - for prisms and cylinders with different radii and heights to see how the volume changes. These explorations can be fun and informative, helping to deepen the understanding of three-dimensional spaces.

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