# Finding Surface Areas of Prisms and Cylinders – An Ultimate Guide

Finding the surface areas of prisms and cylinders is a fundamental concept in geometry and is crucial in various real-world applications, such as construction, architecture, and engineering. These calculations help determine the amount of material needed for covering or painting objects and provide insights into their physical properties.

**Surface Area of Prisms:**

A prism is a three-dimensional geometric shape with two parallel and congruent polygonal bases connected by rectangular or parallelogram faces. To find the surface area of a prism, you need to calculate the areas of its individual components and then sum them up.

**Base Area**: Calculate the area of one of the polygonal bases (usually a rectangle or triangle) by multiplying its length and width. For example, for a rectangular prism, you would multiply the length and width of the rectangular base.**Lateral Surface Area**: Determine the total area of the rectangular or parallelogram faces that connect the two bases. This is done by calculating the perimeter of one base and multiplying it by the height of the prism. For a rectangular prism, it’s the sum of all four sides’ lengths times the height.**Total Surface Area**: Add the base area to the lateral surface area to obtain the total surface area of the prism.

**Surface Area of Cylinders:**

A cylinder consists of two circular bases that are parallel and congruent, connected by a curved surface. Calculating the surface area of a cylinder involves specific formulas:

**Base Area**: Find the area of one of the circular bases by squaring the radius and multiplying it by π (pi). The formula is A = πr², where ‘r’ is the radius.**Lateral Surface Area**: Determine the area of the curved surface, which is essentially a rectangle when “unrolled.” This is calculated by multiplying the circumference of the base by the height of the cylinder. The formula is A = 2πrh, where ‘r’ is the radius and ‘h’ is the height.**Total Surface Area**: Add the base area and the lateral surface area to find the total surface area of the cylinder.

In both cases, understanding the surface area of prisms and cylinders is essential for various practical scenarios. Whether you’re painting a room (prism) or calculating the material needed to create a cylindrical container, these calculations are fundamental for efficient resource planning and accurate project execution. Mastery of these concepts is valuable not only in academic settings but also in everyday problem-solving and professional applications.

## Rectangular Prism

$$ \large \text{Surface Area} = 2 \ell w + 2 \ell h + 2 w h $$

## Circular Cylinder Prism

$$ \large \text{Surface Area} = 2 \pi r^2 + 2 \pi r h $$

## Question 1

Calculate the surface area of a cube of side \( 5 \) cm.

\( \begin{align} \text{Surface Area} &= 6s^2 \\ &= 6 \times 5^2 \\ &= 150 \text{ cm}^2 \end{align} \)

## Question 2

A cube has a surface area of \( 3750 \) cm^{2}.

(a) Find the area of each face.

\( \begin{align} 6s^2 &= 3750 \\ s^2 &= 625 \text{ cm}^2 \end{align} \)

(b) Find the length of each edge.

\( \begin{align} s^2 &= 625 \\ s &= 25 \text{ cm} \end{align} \)

## Question 3

A rectangular swimming pool measuring \( 25 \) m long, \( 15 \) m wide and \( 2 \) m deep is to be painted. Find the surface area of the swimming pool. Remember, no ceiling!

\( \begin{align} 2 \times (15 \times 2) + 2 \times (25 \times 2) + 15 \times 25 = 535 \text{ cm}^2 \end{align} \)

## Question 4

Calculate the surface area of the figure below, correcting the nearest cm^{2}.

\( \begin{align} 2 \times 3.5^2 \pi + 2 \pi \times 3.5 \times 15 &= 406.8362 \cdots \\ &= 407 \text{ cm}^2 \end{align} \)

## Question 5

Find the total surface area.

\( \begin{align} \displaystyle \frac{1}{2} \times 8 \times 12 \times 2 + 15 \times 10 \times 2 + 12 \times 15 = 576 \text{ cm}^2 \end{align} \)

The Surface Area of Prisms and Cylinders Explained with Examples

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