Surface Area Of A Right Cylinder Formulas And Explanation

Jul 9, 2025 by Jeany 58 views

To accurately calculate the surface area of a right cylinder, it's crucial to understand the components that make up its total surface. A right cylinder consists of two circular bases and a curved rectangular surface connecting these bases. The formula for the surface area must account for the area of these three components. Let's delve into the formulas that correctly compute the surface area of a right cylinder, where h represents the height, r is the radius, and BA denotes the base area.

Exploring the Formulas for Surface Area

When determining the surface area of a right cylinder, we consider the areas of its constituent parts: the two circular bases and the lateral surface. The base area (BA) of a single circular base is given by ${\pi r^2}$ , where r is the radius of the circle. Since there are two bases, their combined area is ${2 \pi r^2}$ . The lateral surface, when unrolled, forms a rectangle. The length of this rectangle is the circumference of the base circle, which is ${2 \pi r}$ , and the width is the height h of the cylinder. Therefore, the lateral surface area is ${2 \pi rh}$ .

To find the total surface area, we sum the areas of the two bases and the lateral surface. This gives us the formula:

${ ext{Surface Area} = 2 \pi r^2 + 2 \pi rh }$

Now, let's analyze the options provided to see which ones correctly represent this formula or its equivalent forms.

Option A: $B A + \pi r^2$

This formula is incorrect. It includes the base area (BA), which is ${\pi r^2}$ , and adds another ${\pi r^2}$ . This only accounts for the area of the two bases but omits the lateral surface area. For a complete surface area calculation, we must include the lateral surface area, making this option insufficient.

Option B: $B A + 2 \pi r$

This formula is also incorrect. It starts with the base area (BA), which is ${\pi r^2}$ , and adds ${2 \pi r}$ . This term represents the circumference of the base circle, not the lateral surface area or the area of the second base. The correct formula must include the area of both bases ( ${2 \pi r^2}$ ) and the lateral surface area ( ${2 \pi rh}$ ), which this option fails to do.

Option C: $2 B A + 2 \pi r h$

This formula is correct. It accurately represents the surface area of a right cylinder. Here's why:

2BA: This accounts for the area of both circular bases. Since BA is the base area, which is ${\pi r^2}$ , 2BA is ${2 \pi r^2}$ .
2πrh: This represents the lateral surface area of the cylinder. It is calculated by multiplying the circumference of the base ( ${2 \pi r}$ ) by the height (h) of the cylinder.

By adding these two components, we get the total surface area of the cylinder, making this option a valid formula.

Option D: $2(\pi r^2 + \pi r h)$

This formula is also correct. It is an equivalent form of the standard surface area formula. Let's break it down:

πr²: This is the area of one circular base.
πrh: This is half of the lateral surface area. The lateral surface area is calculated as the circumference of the base ( ${2 \pi r}$ ) times the height (h), which gives ${2 \pi rh}$ . Thus, ${\pi rh}$ is half of this.

When we add ${\pi r^2}$ and ${\pi rh}$ , we are considering the area of one base and half the lateral surface. Multiplying this sum by 2 gives us the total surface area, including both bases and the entire lateral surface. This can be shown mathematically:

${ 2(\pi r^2 + \pi r h) = 2 \pi r^2 + 2 \pi r h }$

This is the same as the standard formula, making option D a correct representation of the surface area of a right cylinder.

Option E: $2 B A+h C$

This formula is correct and provides an alternative, yet accurate, way to calculate the surface area of a right cylinder. Here’s a detailed explanation:

2BA: As previously established, BA represents the base area of one circular base, which is ${\pi r^2}$ . Therefore, 2BA accounts for the area of both circular bases, totaling ${2 \pi r^2}$ .
hC: In this context, h stands for the height of the cylinder, and C represents the circumference of the base. The circumference of a circle is calculated using the formula ${C = 2 \pi r}$ . When we multiply the circumference C by the height h, we are calculating the lateral surface area of the cylinder. The lateral surface area is the area of the curved surface connecting the two circular bases. Thus, hC is equivalent to ${h imes 2 \pi r}$ , which equals ${2 \pi rh}$ .

By adding these two components, we are summing the areas of the two circular bases ( ${2 \pi r^2}$ ) and the lateral surface area ( ${2 \pi rh}$ ). This sum gives us the total surface area of the cylinder.

Mathematically, the formula can be represented as:

${ 2BA + hC = 2(\pi r^2) + h(2 \pi r) = 2 \pi r^2 + 2 \pi rh }$

This is the standard formula for the surface area of a right cylinder, confirming that Option E is indeed a valid and correct representation.

In summary, this formula encapsulates the fundamental concept that the total surface area of a cylinder is the sum of the areas of its bases and its lateral surface. The clarity and simplicity of this formula make it a valuable tool in various mathematical and practical applications.

Conclusion

In conclusion, the formulas that correctly find the surface area of a right cylinder are:

Option C: 2BA + 2πrh
Option D: 2(πr² + πrh)
Option E: 2 B A+h C

These formulas account for both the areas of the circular bases and the lateral surface area, providing an accurate calculation of the total surface area of a right cylinder.

By understanding the composition of a cylinder and how each component contributes to the overall surface area, we can confidently apply these formulas in various mathematical and practical contexts.

In summary, to find the surface area of a right cylinder, it is essential to consider the areas of both its circular bases and its lateral surface. The correct formulas, 2BA + 2πrh, 2(πr² + πrh), and 2 B A+h C, effectively capture these components, ensuring an accurate calculation of the total surface area. Understanding these formulas not only aids in mathematical problem-solving but also enhances comprehension of geometric principles in real-world applications. Remember, the surface area is the sum of the areas of all the surfaces, and in a cylinder, this includes two circular bases and a curved lateral surface. Thus, mastering these formulas provides a solid foundation for further explorations in geometry and related fields.