Find The Total Surface Area Of The Following Prism

Understanding how to find the total surface area of a prism is a fundamental skill in geometry that bridges the gap between two-dimensional shapes and three-dimensional objects. Whether you are a student preparing for an exam, a teacher designing a lesson plan, or a professional needing to calculate material costs for a construction project, mastering this concept allows you to quantify the exact amount of space covering the exterior of a solid figure. The process relies on deconstructing a complex 3D shape into manageable 2D faces, calculating the area of each, and summing them to reach a final answer.

What Is a Prism and Why Does Surface Area Matter?

Before diving into calculations, Make sure you define the object in question. It matters. In real terms, a prism is a polyhedron comprising two parallel, congruent polygonal bases connected by lateral faces that are parallelograms (usually rectangles in right prisms). The name of the prism is derived from the shape of its base—triangular prism, rectangular prism, pentagonal prism, hexagonal prism, and so on Not complicated — just consistent..

And yeah — that's actually more nuanced than it sounds.

The total surface area (TSA) represents the sum of the areas of all faces: the two bases and all lateral faces. Think about it: in real-world applications, TSA determines how much paint is needed to cover a box, how much wrapping paper covers a gift, or the amount of sheet metal required to fabricate a duct. This leads to this differs from the lateral surface area (LSA), which excludes the bases. It is a practical measurement of the "skin" of the object.

It sounds simple, but the gap is usually here.

The Universal Formula Approach

While specific formulas exist for every prism type, a single universal formula applies to all right prisms (where lateral edges are perpendicular to the base):

$TSA = 2 \times B + P \times h$

Where:

• $B$ = Area of one base
• $P$ = Perimeter of the base
• $h$ = Height of the prism (the perpendicular distance between the two bases)

This formula works because the lateral faces of a right prism can be "unwrapped" into a single large rectangle. Because of that, the area of this lateral rectangle is $P \times h$. The height of this rectangle is the prism height ($h$), and the width is the perimeter of the base ($P$). Adding the two bases ($2B$) gives the total surface area Not complicated — just consistent..

Step-by-Step Guide for Common Prisms

Since the prompt references "the following prism" without providing a specific diagram, the most valuable approach is to demonstrate the calculation for the most frequently encountered prism types. You can apply the relevant section below to your specific problem.

1. Rectangular Prism (Cuboid)

This is the most common prism, shaped like a standard box. It has six rectangular faces.

Variables: Length ($l$), Width ($w$), Height ($h$) That's the part that actually makes a difference..

Formula: $TSA = 2(lw + lh + wh)$

Steps:

1. Calculate the area of the three distinct face pairs:
   • Top/Bottom: $l \times w$
   • Front/Back: $l \times h$
   • Left/Right: $w \times h$
2. Sum these three areas.
3. Multiply the sum by 2.

Example: Find the TSA of a rectangular prism with $l = 5\text{ cm}$, $w = 3\text{ cm}$, $h = 4\text{ cm}$ Nothing fancy..

• $lw = 15$
• $lh = 20$
• $wh = 12$
• Sum $= 47$
• $TSA = 2 \times 47 = 94\text{ cm}^2$

2. Triangular Prism

This prism has two triangular bases and three rectangular lateral faces. It is crucial to distinguish between the height of the triangle ($h_b$) and the height (length) of the prism ($H$).

Variables: Base of triangle ($b$), Height of triangle ($h_b$), Side lengths of triangle ($s_1, s_2, s_3$), Height of prism ($H$).

Formula: $TSA = 2 \times (\frac{1}{2} b h_b) + (s_1 + s_2 + s_3) \times H$ Simplified: $TSA = b h_b + P_{triangle} \times H$

Steps:

1. Find the area of one triangular base: $B = \frac{1}{2} \times b \times h_b$.
2. Find the perimeter of the triangular base: $P = s_1 + s_2 + s_3$.
3. Calculate Lateral Surface Area: $LSA = P \times H$.
4. Calculate Total Surface Area: $TSA = 2B + LSA$.

Example: A triangular prism has a base triangle with $b = 6\text{ m}$, $h_b = 4\text{ m}$, sides $5\text{ m}, 5\text{ m}, 6\text{ m}$, and prism height $H = 10\text{ m}$.

• Base Area ($B$) $= 0.5 \times 6 \times 4 = 12\text{ m}^2$.
• Perimeter ($P$) $= 5 + 5 + 6 = 16\text{ m}$.
• $LSA = 16 \times 10 = 160\text{ m}^2$.
• $TSA = 2(12) + 160 = 24 + 160 = 184\text{ m}^2$.

3. Cylinder (Circular Prism)

Technically a cylinder is not a polyhedron, but it functions as a prism with circular bases. The lateral surface "unwraps" into a rectangle.

Variables: Radius ($r$), Height ($h$) Simple, but easy to overlook..

Formula: $TSA = 2\pi r^2 + 2\pi r h$ or $2\pi r(r + h)$

Steps:

1. Calculate area of two circular bases: $2 \times \pi r^2$.
2. Calculate lateral area (circumference $\times$ height): $2\pi r \times h$.
3. Add them together.

4. Regular Polygonal Prisms (Pentagonal, Hexagonal, Octagonal)

For prisms with regular polygon bases (all sides and angles equal), the universal formula $TSA = 2B + Ph$ is highly efficient That's the part that actually makes a difference. That's the whole idea..

Variables: Apothem ($a$), Side length ($s$), Number of sides ($n$), Prism height ($h$) Not complicated — just consistent..

Steps:

1. Perimeter: $P = n \times s$.
2. Base Area: $B = \frac{1}{2} \times a \times P$ (Area of regular polygon $= \frac{1}{2} \times \text{apothem} \times \text{perimeter}$).
3. Lateral Area: $LSA = P \times h$.
4. Total: $TSA = 2B + LSA$.

Example (Hexagonal Prism): Side $s = 4\text{ cm}$, Apothem $a \approx 3.46\text{ cm}$, Height $h = 12\text{ cm}$.

• $P = 6 \times 4 = 24\text{ cm}$.
• $B = 0.5 \times 3.46 \times 24 \approx 41.52\text{ cm}^2$.
• $LSA = 24 \times 12 = 288\text{ cm}^2$.
• $TSA = 2(41.52