How to Calculate the Surface Area of a Cube and a Prism Step by Step
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 Formulas for Cubes and Prisms (With Examples)
1. What is the formula for the surface area of a cube?
The surface area of a cube is calculated using the formula 6a², where 'a' represents the length of one side (edge) of the cube. This formula is fundamental in geometry and mensuration, finding applications in various real-world scenarios involving cubic shapes.
2. What is the formula for the surface area of a rectangular prism?
For a rectangular prism (also known as a cuboid), the total surface area is given by the formula 2(lb + bh + hl), where 'l' is the length, 'b' is the breadth (width), and 'h' is the height of the prism. This formula is crucial for calculating the total surface area of various 3D figures in geometry.
3. What is the difference between lateral and total surface area?
The key difference lies in what areas are included: Total surface area considers all the faces of a 3D figure, while lateral surface area only includes the areas of the vertical sides, excluding the top and bottom faces. This distinction is important in mensuration problems.
4. How do you find the surface area of a 3D cube?
To find the surface area of a cube, first measure the length of one side (edge). Then, square this length (multiply it by itself) and multiply the result by 6. This is because a cube has six identical square faces. The formula is 6a², where 'a' is the length of a side. Understanding this is essential for geometry problems.
5. What is the formula for surface area in 3D?
There isn't one single formula for the surface area of all 3D shapes. The formula depends on the specific shape. For example, cubes use 6a², rectangular prisms use 2(lb + bh + hl), and other shapes have their own unique formulas. Mastering these formulas is vital for success in mensuration.
6. What is the surface area formula for a triangular prism?
The surface area of a triangular prism is calculated by adding the areas of its five faces. This involves finding the area of the two triangular bases and the three rectangular lateral faces. The precise formula will vary depending on the dimensions (base, height, and side lengths) of the triangular prism. This topic is covered under geometry and mensuration.
7. How do I solve surface area questions?
Solving surface area questions involves identifying the 3D shape, recalling the correct formula (e.g., 6a² for a cube, 2(lb + bh + hl) for a rectangular prism), substituting the given dimensions, and performing the calculations accurately. Remember to always state your units (e.g., cm², m²). Practice is key to mastering this aspect of mensuration.
8. What are the formulas for surface area and volume of 3D shapes?
Surface area and volume formulas differ for each 3D shape. Surface area measures the total area of the external surfaces, while volume measures the space enclosed within the shape. For cubes, the surface area is 6a² and volume is a³. For rectangular prisms, surface area is 2(lb + bh + hl) and volume is lbh. These formulas are fundamental in geometry and mensuration.
9. What is the surface area formula of a cube and prism?
The surface area of a cube is 6a² (where 'a' is the side length). For a rectangular prism, it's 2(lb + bh + hl) (l=length, b=breadth, h=height). Triangular prisms require calculating the areas of the triangles and rectangles that make up their faces. Understanding these formulas is crucial for mensuration calculations.
10. How is surface area used in real life?
Surface area has many real-world applications. It’s used to determine the amount of paint needed for a wall (rectangular prism), the material needed to wrap a gift (cube or other shapes), or to calculate the heat loss from a building (various 3D shapes). These applications highlight the practical significance of mensuration in everyday life.
