What is the Total Surface Area of a Solid Cylinder if its Radius and Height are 4 cm and 10 cm Respectively?

The total surface area of a solid cylinder with a radius of 4 cm and a height of 10 cm is 352 square centimeters (cm²).

Understanding Cylinder Surface Area

A solid cylinder comprises two main parts: two identical circular bases and a curved lateral surface that connects them. The total surface area (TSA) is the sum of the areas of these components. Imagine flattening a cylinder; you would get two circles and a rectangle, where the rectangle's length is the circumference of the base and its width is the cylinder's height.

Formula for Total Surface Area (TSA)

The standard formula to calculate the total surface area of a solid cylinder is:

TSA = 2πr(r + h)

Where:

r represents the radius of the cylinder's base.
h denotes the height of the cylinder.
π (pi) is a mathematical constant, commonly approximated as 3.14159 or 22/7 for various calculations.

Step-by-Step Calculation

Let's calculate the total surface area using the given dimensions:

Radius (r): 4 cm
Height (h): 10 cm
For this calculation, we will use the common approximation for pi: π ≈ 22/7.

Substitute the given values into the formula:
TSA = 2 × (22/7) × 4 × (4 + 10)
Simplify the expression inside the parentheses first:
TSA = 2 × (22/7) × 4 × (14)
Multiply the numerical terms. Notice that 14 is a multiple of 7, which simplifies the division:
TSA = (2 × 22 × 4 × 14) / 7
TSA = 2 × 22 × 4 × (14 / 7)
TSA = 2 × 22 × 4 × 2
Perform the final multiplication:
TSA = 44 × 8
TSA = 352

Therefore, the total surface area of the cylinder is 352 cm².

Key Components of the Surface Area

The total surface area can be broken down into the individual areas of its components:

Area of the two circular bases (top and bottom): 2 × πr²
- 2 × (22/7) × (4 cm)² = 2 × (22/7) × 16 cm² = 704/7 cm² ≈ 100.57 cm²
Lateral (Curved) Surface Area: 2πrh
- 2 × (22/7) × 4 cm × 10 cm = 1760/7 cm² ≈ 251.43 cm²

Adding these two components together gives:
704/7 cm² + 1760/7 cm² = 2464/7 cm² = 352 cm². This confirms the total surface area derived from the combined formula.

Summary of Cylinder Dimensions and Area

Parameter	Value	Unit
Radius (`r`)	4	cm
Height (`h`)	10	cm
Pi (`π`) (used)	22/7	(Approximation)
Total Surface Area	352	cm²

Practical Applications of Surface Area Calculations

Calculating the surface area of cylinders is a fundamental skill with broad applications across various fields:

Manufacturing and Material Estimation: Industries use these calculations to determine the amount of raw material needed for producing cylindrical objects like pipes, tanks, and packaging (e.g., soda cans, aerosol containers).
Painting and Coating: Professionals estimate the quantity of paint, anti-corrosion treatments, or insulation required to cover cylindrical structures such as water heaters, pipelines, or columns.
Heat Transfer Engineering: In engineering, the exposed surface area of a cylinder is crucial for designing efficient heat exchangers, radiators, or cooling systems, as it directly impacts heat dissipation or absorption.
Storage and Volume Optimization: Understanding surface area helps in designing containers that optimize the balance between storage capacity and material cost.
Architecture and Construction: Architects and builders consider surface area for cladding, finishing, and structural integrity of cylindrical architectural elements.

Understanding "Exact Answer" in Context

When performing calculations involving π, the term "exact answer" can have two interpretations:

Symbolic Exactness: This refers to leaving π as a symbol in the answer (e.g., 112π cm²). This is the most mathematically precise form, as π is an irrational number with an infinite, non-repeating decimal representation.
Numerical Exactness (with Approximation): In many practical scenarios, and as demonstrated by the calculation method provided, π is approximated (e.g., 22/7 or 3.14). When a specific approximation for π is used, the resulting numerical value (like 352 cm²) is considered "exact" given that particular approximation.

For this question, by using π ≈ 22/7, we arrive at the precise numerical result of 352 cm².

Further Resources:

For more details on the surface area of a cylinder, explore Khan Academy.
Understand general geometry concepts and formulas on Cuemath.