Surface Area and Lateral Area Formulas of Prisms with Solved Examples
The concept of Surface and Lateral Area of 3D Figures: Prisms Formulas is essential in Geometry and Mensuration, especially for students preparing for school exams, competitive exams like JEE, and for those curious about real-world uses such as design and architecture. Understanding these formulas makes it much easier to solve problems related to packaging, construction, and many other daily scenarios.
What is Surface and Lateral Area of Prisms?
A prism is a three-dimensional (3D) solid that has two identical parallel bases and flat faces connecting the bases. Examples include rectangular prisms (boxes), triangular prisms (roof shapes), and hexagonal prisms. The surface area of a prism tells us how much material would be needed to cover it entirely, while the lateral area refers only to the area of the sides, excluding the bases.
At Vedantu, we help you master these concepts so that you can quickly spot which formula to use, understand the meaning of each part of the formula, and apply your knowledge confidently.
Key Terms and Definitions
- Prism: 3D object with two parallel, congruent bases and lateral faces that are parallelograms.
- Base (B): The repeating, parallel face at the top and bottom.
- Height (h): The perpendicular distance between the two bases.
- Lateral faces: The side faces connecting the bases.
- Lateral Surface Area (LSA): Total area of only the side (lateral) faces.
- Total Surface Area (TSA): Sum of all faces’ areas — side faces plus the two bases.
- Perimeter (P): Total distance around the base.
Formulas for Surface Area of Prisms
There are two main formulas related to surface area for prisms:
- Lateral Surface Area (LSA) = Perimeter of base × Height = P × h
- Total Surface Area (TSA) = Lateral Surface Area + 2 × Area of Base = P × h + 2B
| Prism Type | Base Area (B) | Base Perimeter (P) | LSA Formula | TSA Formula |
|---|---|---|---|---|
| Rectangular Prism | l × w | 2(l + w) | 2(l + w) × h | 2(lw + lh + wh) |
| Triangular Prism | ½ × base × height | Sum of three sides of triangle | (Sum of sides) × h | (Sum of sides) × h + 2 × (area of base) |
| Hexagonal Prism | \(\frac{3\sqrt{3}}{2} a^2\) | 6a | 6a × h | 6a × h + 2 × \(\frac{3\sqrt{3}}{2} a^2\) |
Example Problems: Step-by-Step Solutions
Example 1: Rectangular Prism
Find the lateral and total surface area of a rectangular prism with length 8 cm, width 5 cm, and height 4 cm.
- Perimeter of base, P = 2(8 + 5) = 2(13) = 26 cm
- Area of base, B = 8 × 5 = 40 cm²
- Lateral Surface Area = P × h = 26 × 4 = 104 cm²
- Total Surface Area = (P × h) + 2B = 104 + 2 × 40 = 184 cm²
Example 2: Triangular Prism
A triangular prism has a base triangle with sides 3 cm, 4 cm, and 5 cm, and the length (height of prism) is 10 cm.
- Perimeter of base, P = 3 + 4 + 5 = 12 cm
- Area of base, B = ½ × 3 × 4 = 6 cm²
- Lateral Surface Area = P × h = 12 × 10 = 120 cm²
- Total Surface Area = (P × h) + 2B = 120 + 2 × 6 = 132 cm²
Practice Problems
- Find the TSA and LSA of a rectangular prism with l = 6 cm, w = 4 cm, h = 3 cm.
- A triangular prism has a base area of 10 cm², perimeter 12 cm, and height 7 cm. Find its LSA and TSA.
- Find the TSA of a hexagonal prism with base edge a = 2 cm and height h = 5 cm. (Use area formula for hexagon: \(\frac{3\sqrt{3}}{2} a^2\))
- If the base of a prism is a pentagon with area 22 cm², perimeter 15 cm, and the height is 8 cm, calculate the lateral and total surface area.
Common Mistakes to Avoid
- Mixing up lateral area with total surface area (always remember TSA includes the two bases too).
- Using height of the base (triangle) instead of the height of the prism (distance between bases).
- Forgetting to double the base area when calculating TSA.
- Mixing up perimeter and area in the formulas.
- Using inconsistent units — always stick to one unit system in each problem.
Real-World Applications
Surface and lateral area concepts are everywhere! Architects use these calculations to estimate paint or tiles for buildings. Designers compute box surface area for packaging. In science projects, knowing the area of a prism-shaped tank helps decide waterproofing material needed. Even wrapping a birthday present requires knowing the total surface area!
For more on visualizing 3D shapes, see our guide on Three-Dimensional Shapes and Their Properties or on formulas for other shapes like Surface Area of a Cylinder.
At Vedantu, we simplify topics like Surface and Lateral Area of 3D Figures: Prisms so you can focus on building your maths skills for school, entrance exams, and modern careers.
In summary, understanding how to calculate the surface and lateral areas of prisms enables students to solve practical and academic problems confidently. Knowing when to use which formula and how to avoid common mistakes is key for excellent results in exams and real-life applications.
FAQs on Surface and Lateral Area of Three Dimensional Prisms
1. What is the formula for the surface area of a prism?
The surface area of a prism is given by SA = 2B + Ph, where B is the area of the base, P is the perimeter of the base, and h is the height of the prism.
- 2B represents the area of the two parallel bases.
- Ph represents the lateral area (sum of all rectangular faces).
2. What is the formula for the lateral area of a prism?
The lateral area of a prism is calculated using the formula LA = Ph, where P is the perimeter of the base and h is the height.
- Lateral area includes only the side faces.
- It does not include the top and bottom bases.
3. How do you find the surface area of a rectangular prism?
The surface area of a rectangular prism is found using SA = 2(lw + lh + wh), where l is length, w is width, and h is height.
- Find the area of each pair of opposite faces.
- Add them together and multiply by 2.
4. How do you find the lateral area of a rectangular prism?
The lateral area of a rectangular prism is found using LA = 2h(l + w).
- Add the length and width of the base.
- Multiply by 2 and then by the height.
5. What is the difference between lateral area and total surface area of a prism?
The lateral area includes only the side faces, while the total surface area includes both the lateral faces and the two bases.
- Lateral Area = Ph
- Surface Area = 2B + Ph
6. How do you calculate the surface area of a triangular prism?
The surface area of a triangular prism is calculated using SA = 2B + Ph, where B is the area of the triangular base.
- First, find the area of the triangle using ½ × base × height.
- Find the perimeter of the triangle.
- Multiply the perimeter by the prism height.
- Add twice the base area.
7. Can you give an example of finding the surface area of a prism?
Yes, to find the surface area of a prism, use SA = 2B + Ph and substitute the known values.
- Base area B = 15 cm²
- Perimeter P = 16 cm
- Height h = 6 cm
Step 2: Ph = 16 × 6 = 96
Total SA = 30 + 96 = 126 cm².
8. Why is the formula for lateral area of a prism Ph?
The formula LA = Ph works because the lateral faces of a prism form rectangles whose combined width equals the base perimeter.
- Each side of the base forms one rectangular face.
- All rectangles have height h.
- Adding their widths equals the perimeter P.
9. What units are used for surface area and lateral area of prisms?
Surface area and lateral area are measured in square units such as cm², m², or in².
- Length is measured in units (cm, m, in).
- Area is always measured in squared units.
10. What are common mistakes when finding the surface area of a prism?
A common mistake when finding surface area of a prism is forgetting to include both base areas.
- Using LA = Ph but not adding 2B.
- Incorrectly calculating the base area.
- Using the wrong perimeter for irregular bases.
- Forgetting to square the units in the final answer.
