How to Calculate Surface Area of Prisms: Step-by-Step Guide
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 Surface Area of Prisms Made Simple: Formulas, Types & Examples
1. How to find the surface area of prisms?
The surface area of a prism is calculated by summing the areas of all its faces. For a right prism, a common formula is: Surface Area = 2 × Base Area + Perimeter of Base × Height. This formula is adaptable to various prism types, including rectangular, triangular, and other regular prisms.
2. What is the formula of prism?
There isn't one single formula for all prisms, as the shape of the base significantly impacts the calculation. However, the general approach involves finding the area of each face and summing them. For a right prism, the formula Surface Area = 2 × Base Area + Perimeter of Base × Height provides a straightforward calculation. Remember to adjust the base area calculation based on the prism's base shape (e.g., triangle, rectangle).
3. What is the rule for surface area of a prism?
The fundamental rule is to calculate the area of each face of the prism and add them together to find the total surface area. The specific formulas vary based on prism type; however, for a right prism, use Surface Area = 2 × Base Area + Perimeter of Base × Height. Understanding lateral surface area (the area of the sides only) versus total surface area (including bases) is also crucial.
4. What are the formulas for surface area?
Surface area formulas depend on the prism's shape. For a right prism, a common formula is: Surface Area = 2 × Base Area + Perimeter of Base × Height. However, for other prisms (like oblique prisms), more complex calculations are required. The base area and perimeter must be determined separately based on the shape of the prism’s base (e.g., triangle, rectangle, hexagon).
5. What’s the formula for surface area of a triangular prism?
The surface area of a triangular prism is calculated by finding the area of its three rectangular faces and two triangular bases and adding them together. You can also use the formula: Surface Area = 2 × Area(Triangle) + (Perimeter of Triangle) × Height, where the triangle is the prism's base.
6. What’s the formula for surface area of a rectangular prism?
The surface area of a rectangular prism, also known as a cuboid, can be calculated using the formula: Surface Area = 2(lb + bh + lh), where 'l' is length, 'b' is breadth, and 'h' is height. This formula accounts for the areas of the six rectangular faces.
7. What is the lateral surface area of a prism?
Lateral surface area refers to the total area of the lateral faces of a prism—excluding the areas of the bases. For a right prism, it’s simply the perimeter of the base multiplied by the height. This is useful when calculating only the area to be painted on the sides of a prism.
8. How is total surface area different from lateral surface area?
Total surface area includes the areas of all faces of a prism, including the top and bottom bases. Lateral surface area considers only the areas of the sides, excluding the bases. The total surface area formula for a right prism is 2 × Base Area + Perimeter of Base × Height; the lateral surface area is simply Perimeter of Base × Height.
9. What units are surface area answers reported in?
Surface area is always measured in square units because it represents a two-dimensional measurement. Common units include square centimeters (cm²), square meters (m²), square feet (ft²), etc. Always ensure consistent units throughout your calculations.
10. How does the surface area formula change for irregular prism bases?
For prisms with irregular bases, you must calculate the area of the irregular base using appropriate methods (e.g., dividing into smaller shapes). Then, calculate the perimeter of the irregular base. The formula remains similar in concept: Total Surface Area = 2 × (Area of Irregular Base) + (Perimeter of Irregular Base) × Height.
11. Why is it useful to distinguish between lateral and total surface area?
Differentiating between lateral and total surface area is crucial for real-world applications. For example, if you are painting a prism, you'd only need the lateral surface area. However, if you were calculating the material needed for packaging, you'd use the total surface area.
12. Can you derive the surface area formula from the prism net?
Yes, a prism net visually displays all faces of the prism in a 2D layout. By calculating the area of each face on the net and adding them, you can derive the total surface area formula. This method provides a visual understanding and verification of the formula.
