Surface Area Formulas and Step-by-Step Examples
The concept of Finding Surface Area is a key building block in geometry and mensuration. It is crucial for students preparing for school exams, JEE, NEET, and for anyone keen on understanding how much material is required to cover or paint a solid object in real life. At Vedantu, we make learning topics like finding surface area clear and simple for all learners.
Understanding Surface Area
Surface area is the total area that covers the outer surface of a 3D (three-dimensional) object. Unlike area (which covers flat shapes) and volume (which fills space), surface area is all about “wrapping” the outside of solids. Think of wrapping a gift box—the paper covers all outer faces; the total area used is the surface area.
The main difference is:
- Area: For 2D (flat) shapes, like rectangles or circles (e.g., floor size).
- Surface Area: For 3D shapes, like cubes, spheres, and cylinders (e.g., amount of paint needed).
- Volume: Amount of space inside the solid (e.g., water a tank can hold). For more, see Difference between Area and Surface Area.
Surface area is measured in square units, such as cm², m², or in².
Key Surface Area Formulas
Each 3D shape has its own formula for total and lateral (side) surface area. Here is a quick table of the most common solids you’ll see in school and exams:
| Shape | Lateral Surface Area (LSA) | Total Surface Area (TSA) |
|---|---|---|
| Cube | 4a² | 6a² |
| Cuboid | 2h(l + b) | 2(lb + bh + lh) |
| Right prism (e.g., rectangular or triangular) | Base perimeter × Height | LSA + 2 × (Area of Base) |
| Right Circular Cylinder | 2πrh | 2πr(r + h) |
| Right Circular Cone | πrl | πr(l + r) |
| Sphere | – | 4πr² |
| Hemisphere | 2πr² | 3πr² |
| Pyramid | (1/2) × Perimeter of base × Slant height | LSA + Area of base |
LSA is the area of only the sides (no top/base); TSA is the area of all surfaces.
For more on specific shapes, see: Cube, Cuboid, Cone, Surface Area of Cuboid.
Steps to Find Surface Area
To find the surface area of any 3D object, you can use these steps:
- Identify the type of solid (cube, cuboid, cylinder, etc.).
- Write down its surface area formula (see table above).
- Substitute the known dimensions (like length, breadth, height, radius, etc.).
- Calculate and write the unit (usually cm², m², etc.).
Tip: Drawing the net (the unfolded faces) helps you visualize what area to add up. Explore more on nets on Nets of Solid Shapes.
Worked Examples by Shape
Example 1: Surface Area of a Cube
Q: Find the total surface area of a cube with side 4 cm.
- Formula: TSA = 6a²
- Here, a = 4 cm
- TSA = 6 × (4)² = 6 × 16 = 96 cm²
Example 2: Surface Area of a Rectangular Cuboid
Q: A cuboid has length = 5 cm, breadth = 3 cm, height = 2 cm. Find its total surface area.
- Formula: TSA = 2(lb + bh + lh)
- Substitute l = 5, b = 3, h = 2
- TSA = 2[(5×3) + (3×2) + (2×5)] = 2[15 + 6 + 10] = 2[31] = 62 cm²
Learn more: Surface Area of Cuboid
Example 3: Surface Area of a Cylinder
Q: Calculate the curved surface area (CSA) and total surface area (TSA) of a cylinder with radius r = 7 cm, height h = 10 cm.
- CSA = 2πrh = 2 × 3.14 × 7 × 10 = 439.6 cm²
- TSA = 2πr(r + h) = 2 × 3.14 × 7 × (7 + 10) = 2 × 3.14 × 7 × 17 = 747.16 cm²
See practical visualizations: Surface Area of a Cylinder
Example 4: Surface Area of a Sphere
Q: What is the surface area of a sphere with radius 5 cm?
- Formula: SA = 4πr²
- SA = 4 × 3.14 × 5 × 5 = 4 × 3.14 × 25 = 314 cm²
Practice Problems
- Find the total surface area of a cube with side 6 cm.
- A right circular cylinder has radius 5 cm and height 8 cm. Calculate its total surface area.
- The length, breadth, and height of a tank are 12 cm, 8 cm, and 6 cm. Find the total surface area.
- Find the curved surface area of a cone of radius 4 cm and slant height 9 cm.
- A sphere has a radius of 10 cm. What is its surface area?
- Challenge: A triangular prism has base perimeter 12 cm and height 7 cm; its base area is 18 cm². What is its total surface area?
Answers:
1. 6×6² = 216 cm²
2. 2πr(r+h) = 2×3.14×5×(5+8) = 2×3.14×5×13 = 408.2 cm²
3. 2(lb + bh + lh) = 2(96+48+72) = 2×216 = 432 cm²
4. πrl = 3.14×4×9 = 113.04 cm²
5. 4πr² = 4×3.14×100 = 1256 cm²
6. LSA = 12×7 = 84 cm²; TSA = 84 + 2×18 = 120 cm²
Common Mistakes to Avoid
- Mixing up formulas between cube, cuboid, cylinder, etc. Always double-check your solid and formula.
- Forgetting to square or correctly use units, e.g., writing “cm” instead of “cm².”
- Including/excluding bases by mistake (especially for open or closed surfaces).
- Not adding all sides when finding total surface area (particularly in composite solids).
- Swapping radius and height for cylinders and cones.
Real-World Applications
Surface area calculations are everywhere in the real world:
- Painting and packaging: Calculating how much paint is needed, or wrapping paper for gifts/boxes.
- Engineering: Designing and making tanks, pipes, or cans for maximum material efficiency.
- Biology and Chemistry: Studying how cells or objects exchange heat or substances—higher surface area means faster reactions or cooling/warming.
- Construction: Tiling floors, wall coverings, or covering roofs are everyday examples.
- For more on related geometry, see Area of Shapes.
Page Summary
We explored how to find surface area for various 3D shapes using clear formulas and practical examples. Remember: always identify your shape, pick the correct formula, and keep track of units. At Vedantu, we support your maths journey with simple steps, examples, and interactive resources. Understanding surface area helps you solve real problems—at school, in exams, and in daily life.
- Surface Area – More Types and Formulas
- Difference between Area and Surface Area
- Cuboid and Cube – Properties and Surface Area
- Surface Area of Cone
- Surface Area of a Cylinder
- Area of Shapes
- Maths Formulas
FAQs on How to Find Surface Area of 3D Shapes
1. What is the formula for surface area?
The surface area formula depends on the shape of the 3D object. For example, a cube's surface area is 6a2 (where 'a' is the side length), while a cylinder's is 2πr2 + 2πrh (where 'r' is the radius and 'h' is the height). There are different formulas for spheres, cones, prisms, and other shapes. Remember to use the correct formula for the shape you're working with.
2. How do you calculate total surface area of a prism?
Calculating the total surface area of a prism involves finding the area of all its faces and adding them together. This includes the lateral surface area (the area of the sides) and the areas of the two bases. The exact formula varies depending on the shape of the base (triangular prism, rectangular prism, etc.). You need to know the dimensions (length, width, height, base area etc.) of the prism to substitute into the appropriate formula.
3. What’s the difference between area, surface area, and volume?
Area measures the space inside a 2D shape (like a square or circle). Surface area measures the total area of the outer surfaces of a 3D object (like a cube or sphere). Volume measures the amount of space inside a 3D object.
4. How can I find the surface area if only given some dimensions?
If you're only given some dimensions, you might need to use other information or geometric relationships to find the missing ones. For instance, if you're given the diagonal of a cube, you can calculate its side length using the Pythagorean theorem and then use the surface area formula. Drawing a diagram and using known geometric properties can help solve for missing values.
5. Where is surface area used in real life?
Surface area calculations are vital in many real-world applications. Examples include determining the amount of paint needed for a wall (area), calculating the material required for packaging (surface area), or designing efficient heat exchangers (surface-to-volume ratio).
6. What is the formula for SA V?
There's no single formula for 'SA V'. 'SA' typically refers to surface area, and 'V' refers to volume. The relationship between surface area and volume is shape-dependent. For example, the surface-to-volume ratio of a sphere is different from that of a cube. The formulas for both surface area and volume must be calculated individually for a given shape.
7. How to calculate total surface area?
To calculate the total surface area, first identify the shape of the object. Then, use the appropriate formula for that shape. Make sure you have all the necessary dimensions (length, width, height, radius, etc.). Substitute these values into the formula and calculate the result. Remember to use correct units (e.g., cm2, m2).
8. How to calculate specific surface area?
Specific surface area is usually related to materials science and refers to the surface area per unit mass or volume of a material. The calculation depends heavily on the material's structure and porosity. Specific formulas are not universal and would need to be tailored to the specific material properties. You might need to consider techniques like microscopy or gas adsorption to determine surface area.
9. Finding surface area of a triangular prism
The surface area of a triangular prism is calculated by summing the areas of its five faces: two triangular bases and three rectangular lateral faces. You will need the dimensions of the triangle (base and height) and the prism's length. The formula involves calculating the area of the triangles and the rectangles and then adding them together.
10. Finding surface area of a cylinder
The surface area of a cylinder is calculated using the formula 2πr2 + 2πrh, where 'r' is the radius and 'h' is the height. The 2πr2 represents the area of the two circular bases, and 2πrh represents the lateral surface area. Remember to use the correct units for radius and height to get the area in square units.
11. Finding surface area of a cube
The surface area of a cube is simply 6 times the square of its side length (a). So the formula is 6a2. If the side length of the cube is 5cm, for example, its surface area would be 6 * 52 = 150 cm2.
