Surface Area Formulas for Prisms and Cubes with Solved Examples
The Surface Area of 3D Figures Prisms and Cubes Formulas are among the most essential concepts in geometry, especially for school students preparing for exams like CBSE, ICSE, and also for competitive tests. Understanding how to calculate the surface area is vital for grasping 3D object properties, solving real-world measurement problems, and excelling in mathematics. At Vedantu, we make these concepts easy with clear definitions, formulas, and worked examples.
What is the Surface Area of 3D Figures?
The surface area is the total area that the surface of a three-dimensional object occupies. For shapes like cubes and prisms, this means the sum of the areas of all their faces. Surface area is measured in square units (such as cm2 or m2). Knowing surface area helps in tasks such as painting, wrapping, or building objects.
- Cube: A special type of prism with 6 equal square faces.
- Rectangular Prism (Cuboid): A box-shaped figure with 6 rectangular faces.
Formulas for Surface Area of Cubes and Prisms
Here are the key formulas you'll need to find the surface area of commonly encountered 3D figures:
| Shape | Total Surface Area Formula | Lateral Surface Area Formula |
|---|---|---|
| Cube | 6a2 | 4a2 |
| Rectangular Prism (Cuboid) | 2(lb + bh + hl) | 2h(l + b) |
| Triangular Prism | (Perimeter of base × height) + 2 × (Base Area) | Perimeter of base × height |
Where:
- a = edge of cube
- l = length, b = breadth, h = height
Remember, lateral surface area means area of all sides excluding the top and bottom (bases) while total surface area includes every face.
Understanding Surface Area with Visuals
Imagine unfolding a cube into a flat "net"—the 6 squares you see sum up to its total surface area. Similarly, the net of a prism shows each face as a rectangle or other polygon. This can help students visually count and calculate each area properly. (For diagrams, refer to your textbook or see Vedantu's Nets of Solid Shapes page.)
Worked Examples
Let’s look at practical examples to see how these formulas work:
Example 1: Surface Area of a Cube
Find the total surface area of a cube with side length 5 cm.
- Total Surface Area = 6a2 = 6 × (5)2 = 6 × 25 = 150 cm2
Example 2: Surface Area of a Rectangular Prism
Find the total surface area of a box measuring 10 cm long, 6 cm wide, and 4 cm tall.
- Total Surface Area = 2(lb + bh + hl)
- = 2[(10×6) + (6×4) + (4×10)]
- = 2[60 + 24 + 40] = 2 × 124 = 248 cm2
Example 3: Lateral Surface Area of a Cube
Side of cube = 8 cm.
Lateral Surface Area = 4a2 = 4 × (8)2 = 4 × 64 = 256 cm2
Practice Problems
- Calculate the total surface area of a cube with each side 7 cm.
- A rectangular prism has length 12 cm, width 8 cm, height 5 cm. Find its total surface area.
- Find the lateral surface area of a cube with side 10 cm.
- The base of a prism is a rectangle of 4 cm by 6 cm, and its height is 9 cm. Find its lateral surface area.
- If the surface area of a cube is 486 cm2, find the length of its side.
Common Mistakes to Avoid
- Forgetting that all faces of the cube are squares, not rectangles.
- Not squaring the edge when using 6a2 for a cube.
- Using the wrong units—always express area in square units.
- Confusing lateral surface area with total surface area (lateral excludes top and bottom).
- Mixing up length, breadth, and height on a cuboid.
Real-World Applications
Surface area calculations are essential in real life. For example, knowing the surface area helps you determine how much paint to buy for a box-shaped tank, how much glass is needed for a fish aquarium, or how much wrapping paper is needed for gifts. Engineers, architects, and designers regularly use these formulas in their work. At Vedantu, these connections are made clear to ensure a practical understanding for every student.
Page Summary
In summary, mastering the surface area of 3D figures prisms and cubes formulas is key to excelling in school mathematics and applying knowledge to real-life problems. Take time to understand the differences between lateral and total surface area, practice with a variety of questions, and avoid common errors. For more help on surface area, check out Surface Area of Cuboid or download formula sheets from Vedantu’s resources section.
FAQs on Surface Area of 3D Figures for Prisms and Cubes Explained
1. What is the formula for the surface area of a prism?
The surface area of a prism is given by the formula 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.
- B = area of one base
- P = perimeter of the base
- h = height (distance between the two bases)
2. What is the surface area formula for a cube?
The surface area of a cube is calculated using the formula SA = 6a², where a is the length of one side.
- A cube has 6 equal square faces.
- Area of one face = a²
- Total surface area = 6 × a²
3. How do you calculate the surface area of a rectangular prism?
The surface area of a rectangular prism is found using SA = 2(lw + lh + wh), where l, w, and h are length, width, and height.
- Find the area of each pair of opposite faces.
- Add them together.
- Multiply the total by 2.
4. What is the difference between lateral surface area and total surface area of a prism?
The lateral surface area of a prism includes only the side faces, while the total surface area includes the side faces plus the two bases.
- Lateral Surface Area = Ph
- Total Surface Area = 2B + Ph
5. How do you find 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.
- Find area of triangle: B = ½ × base × height
- Find perimeter (P) of the triangle
- Multiply P by prism height (h)
- Add 2B + Ph
6. Why is the surface area of a cube 6a²?
The surface area of a cube is 6a² because a cube has six identical square faces, each with area a².
- Area of one face = a × a = a²
- Number of faces = 6
- Total surface area = 6 × a²
7. Can you give an example of finding the surface area of a cube?
To find the surface area of a cube with side length 7 cm, use SA = 6a².
- a = 7 cm
- a² = 49
- SA = 6 × 49 = 294 cm²
8. What units are used for surface area of 3D figures?
The surface area of 3D figures is measured in square units, such as cm², m², or in².
- If dimensions are in centimeters, answer is in cm².
- If dimensions are in meters, answer is in m².
9. How is surface area different from volume in prisms and cubes?
Surface area measures the total area of all outer faces, while volume measures the space inside a 3D figure.
- Surface Area → measured in square units (e.g., cm²)
- Volume → measured in cubic units (e.g., cm³)
10. What are common mistakes when finding the surface area of prisms?
Common mistakes when calculating surface area of prisms include forgetting a face or using the wrong formula.
- Not adding both bases in total surface area.
- Confusing lateral surface area with total surface area.
- Using incorrect perimeter in the formula SA = 2B + Ph.
- Forgetting to square the units.
