Surface Area of Prisms: Formulas and Examples

📚 Quick Study Guide: Surface Area of Prisms

• 📐 What is a Prism? A prism is a three-dimensional geometric shape with two identical, parallel bases and rectangular (or parallelogram) lateral faces connecting them. The shape of the base defines the type of prism (e.g., triangular prism, rectangular prism).
• 🔍 Lateral Surface Area (LSA): This is the sum of the areas of all the lateral faces (the 'sides' of the prism), excluding the two bases. It's like wrapping paper around the sides of a gift box, but not the top or bottom.
• 🔢 LSA Formula: The general formula for the Lateral Surface Area of any prism is $LSA = Ph$, where $P$ is the perimeter of the base and $h$ is the height of the prism (the distance between the two bases).
• ✨ Total Surface Area (TSA): This is the sum of the areas of all faces of the prism, including both bases and all lateral faces. It's the total amount of material needed to construct the entire prism.
• ➕ TSA Formula: The general formula for the Total Surface Area of any prism is $TSA = LSA + 2B$, or $TSA = Ph + 2B$, where $B$ is the area of one base of the prism.
• 📦 Rectangular Prism Formulas:
  • 📏 Perimeter of Base ($P$): $P = 2(l + w)$ (where $l$ is length, $w$ is width)
  • 🖼️ Area of Base ($B$): $B = lw$
  • 🔺 Lateral Surface Area ($LSA$): $LSA = 2(l + w)h$
  • 🌐 Total Surface Area ($TSA$): $TSA = 2(lw + lh + wh)$
• 🧊 Cube Formulas: A cube is a special type of rectangular prism where all sides are equal ($l = w = h = s$).
  • ✅ Lateral Surface Area ($LSA$): $LSA = 4s^2$
  • 🌟 Total Surface Area ($TSA$): $TSA = 6s^2$
• 📐 Triangular Prism Formulas: The base is a triangle. You'll need to calculate the perimeter of the triangular base ($P$) and its area ($B = \frac{1}{2} \text{base}_{triangle} \times \text{height}_{triangle}$) before applying the general LSA and TSA formulas.
• Units: Remember that surface area is always measured in square units (e.g., $cm^2$, $m^2$, $in^2$).

📝 Practice Quiz

1. What is the Lateral Surface Area (LSA) of a prism?

1. The area of the two bases combined.
2. The sum of the areas of all faces, including the bases.
3. The sum of the areas of only the rectangular lateral faces.
4. The volume of the prism.

2. A rectangular prism has a length of 8 cm, a width of 3 cm, and a height of 5 cm. What is its Total Surface Area (TSA)?

1. $120 \text{ cm}^2$
2. $158 \text{ cm}^2$
3. $148 \text{ cm}^2$
4. $94 \text{ cm}^2$

3. Which formula represents the Total Surface Area (TSA) of a cube with side length '$s$'?

1. $s^3$
2. $4s^2$
3. $6s^2$
4. $s^2$

4. A triangular prism has a base that is a right triangle with legs of 3 inches and 4 inches, and a hypotenuse of 5 inches. The height of the prism is 10 inches. What is the Lateral Surface Area (LSA) of this prism?

1. $60 \text{ in}^2$
2. $120 \text{ in}^2$
3. $30 \text{ in}^2$
4. $180 \text{ in}^2$

5. The general formula for the Total Surface Area (TSA) of any prism is:

1. $TSA = P \cdot h$
2. $TSA = P \cdot h + B$
3. $TSA = P \cdot h + 2B$
4. $TSA = \frac{1}{3} B \cdot h$

6. A cube has a total surface area of $294 \text{ m}^2$. What is the length of one of its sides?

1. $6 \text{ m}$
2. $7 \text{ m}$
3. $8 \text{ m}$
4. $9 \text{ m}$

7. For a rectangular prism, if the length, width, and height are all doubled, how does the Total Surface Area change?

1. It doubles.
2. It triples.
3. It quadruples.
4. It remains the same.