Mensuration of Solid Shapes: Cuboids and Cylinders

In our previous studies, we explored flat two-dimensional (2D) shapes like rectangles and circles, focusing on their perimeter and area. In the physical world, however, objects have depth and occupy space. Mensuration of solid shapes allows us to calculate the space inside these three-dimensional (3D) objects (Volume) and the total area of their outer boundaries (Surface Area). This note focuses on two fundamental solid shapes: the cuboid and the cylinder.

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Core Concepts

The Cuboid

A cuboid is a three-dimensional solid bounded by six rectangular faces. It features 12 edges and 8 vertices. The three primary dimensions of a cuboid are its length ($l$), width ($w$), and height ($h$). When all six faces are equal squares, the cuboid is classified as a cube.

To find the capacity or Volume ($V$) of a cuboid, we calculate the area of the rectangular base and multiply it by the height. Because volume measures 3D space, its units are always cubic (e.g., $\text{cm}^3$ or $\text{m}^3$):

$\text{Volume} = \text{length} \times \text{width} \times \text{height} = l \times w \times h$

The Total Surface Area (TSA) is the sum of the areas of all six rectangular faces. Since opposite faces are identical in a cuboid, we group them into three pairs to simplify calculations. Surface area is a 2D measure of the boundary, so its units are square (e.g., $\text{cm}^2$):

$\text{TSA} = 2(lw + lh + wh)$

Worked Example 1

Calculate the Volume and TSA of a cardboard shoebox with a length of $8\text{ cm}$, a width of $5\text{ cm}$, and a height of $3\text{ cm}$.

Solution:

• Volume: $V = l \times w \times h$ $V = 8\text{ cm} \times 5\text{ cm} \times 3\text{ cm} = 120\text{ cm}^3$

• Total Surface Area: $\text{TSA} = 2(lw + lh + wh)$ $\text{TSA} = 2(8 \times 5 + 8 \times 3 + 5 \times 3)$ $\text{TSA} = 2(40 + 24 + 15) = 2(79) = 158\text{ cm}^2$

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The Cylinder

A cylinder is a solid shape with two parallel circular bases connected by a curved surface. If you unroll the curved surface of a cylinder, it forms a flat rectangle. The width of this rectangle corresponds to the height ($h$) of the cylinder, and its length corresponds to the circumference of the circular base ($2\pi r$).

The Volume ($V$) of a cylinder is found by multiplying the area of its circular base ($\pi r^2$) by its height ($h$):

$\text{Volume} = \pi r^2 h$

The surface area of a cylinder is divided into two distinct components:

1. Curved Surface Area (CSA): The area of the unrolled rectangular middle section. $\text{CSA} = 2\pi rh$
2. Total Surface Area (TSA): The sum of the curved surface area and the areas of the two circular ends. $\text{TSA} = 2\pi r^2 + 2\pi rh = 2\pi r(r + h)$

(Note: We typically use $\pi \approx \frac{22}{7}$ or $3.142$ for calculations).

Worked Example 2

Find the Volume and TSA of a closed solid cylindrical tin with a radius of $7\text{ cm}$ and a height of $10\text{ cm}$. (Take $\pi = \frac{22}{7}$).

Solution:

• Volume: $V = \pi r^2 h$ $V = \frac{22}{7} \times 7\text{ cm} \times 7\text{ cm} \times 10\text{ cm}$ $V = 22 \times 7 \times 10 = 1540\text{ cm}^3$

• Total Surface Area: $\text{TSA} = 2\pi r(r + h)$ $\text{TSA} = 2 \times \frac{22}{7} \times 7 \times (7 + 10)$ $\text{TSA} = 44 \times 17 = 748\text{ cm}^2$

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Real-World Scenarios

Understanding the properties of 3D solids is essential in manufacturing, packaging, and construction. For instance, food processing companies use volume calculations to ensure that beverage cans hold the exact amount of liquid advertised on the label without overflowing during assembly.

In construction, civil engineers use surface area formulas to determine the exact amount of anti-corrosive paint needed to coat cylindrical bridge pillars, which prevents material waste and keeps projects within budget.

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Practical Applications

Civil engineers use these geometric formulas to design large municipal water reservoirs, ensuring communities have access to a reliable daily supply. For a quick home-practice activity, find any cylindrical tin of food in your pantry, measure its height and diameter with a ruler, and calculate its volume to see how closely your mathematical result matches the net weight or volume printed on the label.

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Student Reflection Questions

1. If you double the height of a cylinder while keeping its radius constant, what happens to its volume? What happens to the volume if you double the radius instead while keeping the height constant? Explain your reasoning mathematically.
2. A cylindrical pipe is open at both ends. How must you modify the Total Surface Area formula of a closed cylinder to calculate only the exposed metal surface of this open pipe?
3. Imagine you have a lump of clay shaped as a cuboid. If you reshape it entirely into a cylinder without adding or removing any clay, which property remains exactly the same: the Volume or the Total Surface Area? Why?