Volume & Surface Area Of Regular Objects
In this lesson, we will first learn how to identify the base area, also known as the cross-sectional area of regular objects by examining cylinders and prisms. Using the base area, students will then learn how to derive the volume and surface area of regular objects using a general formula, instead of memorising separate formulas for every shape.
Related Lessons:
How To Identify Base Area
Identifying the base area reduces the need to memorise many formulas and builds stronger conceptual understanding.
To determine the base area, when the object is sliced into multiple pieces, and every piece is identical with the same area, that area is the base area, also known as the cross-sectional area.
For example, if we have a cylinder, the base area will be a circle. If we have a prism, the base area will be a triangle. And if we have a cuboid, the base area will be a square or rectangle.
General Volume Formula
The general volume formula is:
Volume = Base Area × Height
where the height is always perpendicular to the base area.
For example, if we have a cylinder radius (r) and height (h):
Volume of cylinder = \(𝜋𝑟^2×ℎ\)
For a prism with triangle base (b), height (h), and length (L):
Volume of prism = \(\frac{1}{2} 𝑏ℎ×𝐿\)
Curved Surface Area Formula
Similarly, we can also make use of the base area to find the curved surface area of the sides of a regular object. The formula is perimeter of the base area, multiply by the height.
For a cylinder,
curved curve S.A. = \(2𝜋𝑟×ℎ\)
For prism with triangle with length of side a, b and c:
curved curve S.A. = \((𝑎+𝑏+𝑐)×𝐿\)
Example 1
The figure shows a closed container of uniform cross-section. The cross section consists of a square ABCD and a quadrant CED of a circle, centre D. Given that AB=7cm, AD=7cm and BH=CG=EF=AI=20cm, calculate
- The area of the cross section ADECB
- The volume of the container
- The surface area BCEFGH
[Take \(𝜋=\frac{22}{7}\)]
Answer:
a) 87.485cm2
b) 1749.7cm3
c) 360cm2
Example 2
A solid cylinder is made from melted iron. The height and diameter of the cylinder measured are 16cm and 15cm respectively.
- Calculate the volume of iron needed to make the cylinder. (Take \(π=3.14\)).
- Ten such cylinders are melted to make cubes whose side is 3mm each. What is the maximum number of cubes that can be made from the cylinders?
Answers:
a) 2826cm3
b) 1,046,666
Watch: Full Concept Breakdown
For students who are still finding the concept difficult to understand, the video below provides a complete explanation of the concepts discussed above. It is especially helpful for students learning the topic for the first time, and also serves as a useful guide for those who face challenges in understanding the concept of base area.