Surface Area of Prisms Formula with Step by Step Solved Examples
The concept of Surface Area of Prisms is an essential part of geometry and mensuration, frequently encountered in school exams, competitive tests like JEE or NEET, and in various real-world situations such as packaging or construction. Mastering surface area calculations for prisms helps students solve problems related to 3D shapes efficiently.
Understanding Surface Area of Prisms
A prism is a three-dimensional solid with two parallel, congruent bases and rectangular lateral faces. The surface area of a prism refers to the total area covered by all its faces, including both bases and the sides (lateral faces). Understanding the difference between lateral surface area (only the sides) and total surface area (sides plus bases) is essential in geometry and mensuration.
Different types of prisms—such as rectangular, triangular, and hexagonal prisms—have slightly different surface area calculations based on the shape of their bases. However, the overall approach to calculating surface area remains consistent:
- Find the area of each base
- Calculate the area of the lateral (side) faces
- Add them to get the total surface area
Formulae for Surface Area of Prisms
The general formula for the surface area of a right prism is:
- Surface Area = 2 × (Base Area) + (Perimeter of Base × Height)
| Prism Type | Formula |
|---|---|
| Rectangular Prism | 2(lb + bh + lh) where l = length, b = breadth, h = height |
| Triangular Prism | (Base Area × 2) + (Perimeter of Triangle × Height) |
| Hexagonal Prism | (2 × Base Area) + (Perimeter of Base × Height) |
In these formulas:
- Base Area: Area of the prism's base (depends on the shape—rectangle, triangle, hexagon, etc.)
- Perimeter of Base: Total length around the base's shape
- Height: The perpendicular distance between the two bases
At Vedantu, we simplify complex formulas like these, giving students easy stepwise methods for all types of prisms.
Worked Examples
Let’s solve some examples to understand the process:
Example 1: Rectangular Prism
Find the surface area of a rectangular prism with length 8 cm, width 5 cm, and height 4 cm.
- Surface Area = 2(lb + bh + lh)
- = 2[(8 × 5) + (5 × 4) + (8 × 4)]
- = 2[40 + 20 + 32]
- = 2[92] = 184 cm²
Example 2: Triangular Prism
A triangular prism has a base triangle with sides 3 cm, 4 cm, 5 cm and height (prism height) 10 cm. Find the surface area.
- Find area of the base (triangle)
Area = ½ × base × height of triangle (Suppose triangle height = 4 cm to side 3 cm as base)
Area = ½ × 3 × 4 = 6 cm² - Perimeter of base = 3 + 4 + 5 = 12 cm
- Surface Area = 2 × 6 + (12 × 10) = 12 + 120 = 132 cm²
Example 3: Hexagonal Prism
A hexagonal prism has a regular base with side 6 cm and height 12 cm. Find the surface area.
- Base Area (hexagon) = (3√3 / 2) × (side)² = (3√3 / 2) × 36 ≈ 93.53 cm²
- Perimeter = 6 × 6 = 36 cm
- Surface Area = 2 × 93.53 + (36 × 12) = 187.06 + 432 = 619.06 cm²
Practice Problems
- Find the surface area of a prism with rectangular base (length 10 cm, width 7 cm, height 5 cm).
- A triangular prism has a base with sides 5 cm, 5 cm, 8 cm, and a height of 15 cm. Find its surface area.
- Calculate the surface area of a pentagonal prism if the base side is 4 cm, prism height is 20 cm (use pentagon base area formula).
- A hexagonal prism has a base side of 8 cm and height 10 cm. Find its total surface area.
- A right prism has a square base of side 6 cm and a height of 9 cm. What is its surface area?
Common Mistakes to Avoid
- Mixing up lateral surface area (just the sides) with total surface area (sides and bases).
- Using the wrong base area formula for different prism shapes.
- Forgetting to multiply the base area by 2 (two bases in a prism).
- Using base height instead of prism height in the formula.
- Reporting answers in linear units (like cm) instead of square units (cm²).
Real-World Applications
Prism surface area calculations are widely used in real life. Architects use them to estimate material needed to build glass buildings (prisms). Manufacturers calculate surface area to determine the amount of wrapping or covering for boxes (cartons are rectangular prisms). Engineers need this concept when painting or creating 3D objects for industries. At Vedantu, our examples and worksheets help students see these practical uses and connect math to everyday contexts.
For more on 3D geometry, explore topics like Surface Area of Cube, Volume of Cube, Cuboid and Cylinder, and Surface Areas and Volumes here at Vedantu.
In this topic, you learned how to find the surface area of prisms, the necessary formulas for various prism types, and saw step-by-step examples. Mastery of this concept ensures you can confidently tackle geometry questions in school, board, and competitive exams, and apply maths skills in real-world problem solving.
FAQs on Understanding the Surface Area of Prisms
1. What is the surface area of a prism?
The surface area of a prism is the total area of all its faces, including the two bases and the lateral faces. In simple terms, it measures how much space the outside of the prism covers.
- It includes 2 base areas.
- It includes the lateral surface area (rectangular faces around the sides).
- It is measured in square units such as cm² or m².
2. What is the formula for the surface area of a prism?
The formula for the surface area of a prism is SA = 2B + Ph, where B is the base area, P is the perimeter of the base, and h is the height.
- 2B represents the two identical bases.
- Ph represents the lateral surface area.
- This formula works for any type of prism (triangular, rectangular, etc.).
3. How do you find the surface area of a rectangular prism?
The surface area of a rectangular prism is calculated using SA = 2(lw + lh + wh), where l = length, w = width, and h = height.
- Step 1: Find areas of all three pairs of faces: lw, lh, wh.
- Step 2: Add them together.
- Step 3: Multiply the sum by 2.
- Example: If l=4, w=3, h=2 → SA = 2(12 + 8 + 6) = 2(26) = 52 square units.
4. How do you calculate the surface area of a triangular prism?
The surface area of a triangular prism is found using SA = 2B + Ph, where B is the area of the triangular base.
- Step 1: Find the area of the triangle (½ × base × height).
- Step 2: Multiply by 2 for both triangular bases.
- Step 3: Find the perimeter of the triangle and multiply by prism height.
- Add all values to get total surface area.
5. What is the lateral surface area of a prism?
The lateral surface area of a prism is the area of all side faces excluding the two bases. It is calculated using LSA = Ph.
- P = perimeter of the base.
- h = height of the prism.
- It represents the "side covering" of the prism.
6. What is the difference between surface area and lateral surface area of a prism?
The surface area includes all faces of a prism, while the lateral surface area includes only the side faces.
- Surface Area = 2B + Ph.
- Lateral Surface Area = Ph.
- Surface area includes bases; lateral surface area does not.
7. Can you give an example of finding the surface area of a prism?
Yes, for a rectangular prism with length 5 cm, width 4 cm, and height 3 cm, the surface area is 94 cm².
- Use SA = 2(lw + lh + wh).
- Calculate: 2(20 + 15 + 12).
- 2(47) = 94 cm².
8. What units are used for the surface area of a prism?
The surface area of a prism is measured in square units.
- Examples: cm², m², in², ft².
- Always square the unit because area measures two-dimensional space.
- If dimensions are in meters, the answer must be in m².
9. Why do we multiply the base area by 2 in the surface area formula?
We multiply the base area by 2 because a prism has two identical parallel bases.
- One base is at the top.
- The other base is at the bottom.
- So total base area contribution is 2B.
10. What are common mistakes when finding the surface area of a prism?
Common mistakes when calculating the surface area of a prism include forgetting faces or using incorrect formulas.
- Forgetting to include both bases.
- Confusing lateral surface area with total surface area.
- Using wrong units (not squaring them).
- Incorrectly calculating the base area.
