What is Surface Area and Volume? (Definition, Formula Table & Solved Questions)
The concept of Surface Area and Volume plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding these measurements helps you work with three-dimensional (3D) objects—such as cubes, cylinders, cones, and spheres—and is essential for success in school exams, entrance tests, and practical life.
What Is Surface Area and Volume?
Surface Area and Volume are two fundamental measurements in geometry. Surface area is the total area covering the outside of a 3D object, while volume measures the amount of space it occupies. You’ll find this concept applied in solid geometry, mensuration, and 3D shape problems at all school levels.
Key Formula for Surface Area and Volume
Here’s a quick reference guide for key shapes:
| Shape | Surface Area | Volume |
|---|---|---|
| Cube | 6a² | a³ |
| Cuboid | 2(lb + bh + hl) | l × b × h |
| Cylinder | 2πr(h + r) | πr²h |
| Sphere | 4πr² | (4/3)πr³ |
| Cone | πr(r + l) | (1/3)πr²h |
Where a is the side, l = length, b = breadth, h = height, r = radius, l (in cone) = slant height.
How to Solve Surface Area and Volume Questions
Solving these problems involves three simple steps:
1. Identify the type of 3D shape and note all given dimensions.2. Choose and apply the correct formula from the table above.
3. Calculate using the right units, and always double-check for any surface(s) that don't count (like hidden or joined surfaces in composite solids).
Example: Find the total surface area and volume of a cuboid with l = 8 cm, b = 5 cm, h = 3 cm.
1. Surface Area = 2(lb + bh + hl) = 2(8×5 + 5×3 + 3×8) = 2(40 + 15 + 24) = 2(79) = 158 cm²2. Volume = l × b × h = 8 × 5 × 3 = 120 cm³
Visual Representation of Surface Area and Volume
Visualizing helps! For example, imagine wrapping a gift box with paper—the wrapper covers the surface area. Filling the box with candies gives you the volume. Diagrams and nets of cubes, cylinders, cones, and spheres are must-see for quick understanding. See more at Surface Area of Cylinder or Volume of Cube, Cuboid and Cylinder.
Difference Between Surface Area and Volume
| Aspect | Surface Area | Volume |
|---|---|---|
| Meaning | Total outside area | Space inside |
| Unit | cm², m² | cm³, m³ |
| Example | Wrapping paper for a box (outer cover) | Water a tank can hold (capacity) |
Common Application Word Problems
Let's try an application:
Question: A cylindrical water tank is 1.5 m in diameter and 2 m in height. Find its surface area and volume.
1. r = diameter/2 = 1.5/2 = 0.75 m; h = 2 m2. Surface Area = 2πr(h + r) = 2 × 3.14 × 0.75 × (2 + 0.75) = 2 × 3.14 × 0.75 × 2.75 ≈ 12.94 m²
3. Volume = πr²h = 3.14 × (0.75)² × 2 = 3.14 × 0.5625 × 2 ≈ 3.53 m³
Surface Area and Volume Calculator Tip
To do calculations faster, use certified online calculators. For instant solutions, try the Surface Area of Rectangular Prism Calculator or Volume of Cuboid Calculator. Always enter units correctly!
Practice Worksheets
Want more practice? Get formula lists at Mensuration Formulas Class 10. Practicing different shapes builds confidence and sharpens speed!
Speed Trick or Vedic Shortcut
To quickly estimate the surface area or volume, round off numbers for mental math, then adjust for exact calculations. For example, approximate π as 3.14 and round dimensions for fast results during exams.
In competitive exams, surface area and volume questions are often "twisted"—like combining a cone and a hemisphere (ice cream model). Just add or subtract corresponding formulas!
Try These Yourself
- Find the total surface area of a cube whose side is 4 cm.
- Calculate the volume of a cylinder with height 10 cm and radius 3 cm.
- What is the difference between the surface area and volume of a sphere of radius 5 cm?
- Solve: A cuboid has l = 7 cm, b = 2 cm, h = 3 cm. Find both surface area and volume.
Frequent Errors and Misunderstandings
- Mixing up formulas for different shapes
- Forgetting to convert all measurements to the same units
- Missing out hidden or joined surfaces in combination or composite solids
- Inserting wrong values for π (use 22/7 or 3.14)
Relation to Other Concepts
Understanding surface area and volume helps with topics like Area and Perimeter, Volume of Cube, Cuboid and Cylinder, and Area and Volume of Solid Shapes. These are direct prerequisites for 3D geometry and advanced mensuration.
Classroom Tip
Remember: Area is the "cover," Volume is the "filling." Try drawing nets of solids and labeling their faces/edges as an activity. Vedantu’s live classes use such diagrams and tricks to help you visualize and learn fast.
We explored Surface Area and Volume—from definition, formula tables, stepwise examples, differences, tricks, and connections to other maths topics. Practice with Vedantu for more diagrams, calculators, and live teaching to become a master in surface area and volume questions.
Practice Questions
Q1: What is the difference between surface area and volume?
Q2: What is the formula for surface area of a cylinder?
Q3: How do I calculate volume of a sphere?
Q4: Why are surface area and volume important in Maths?
Q5: How can I remember all the formulas easily?
Suggested Internal Links
FAQs on Surface Area and Volume in Maths: Concepts, Formulas & Examples
1. What is the fundamental difference between surface area and volume?
The fundamental difference lies in what they measure. Surface area is a two-dimensional measurement of the total area covering the exterior of a three-dimensional object, like the amount of wrapping paper needed for a gift. It is measured in square units (e.g., cm², m²). In contrast, volume is a three-dimensional measurement of the space an object occupies, like the amount of water a bottle can hold. It is measured in cubic units (e.g., cm³, m³).
2. What are some real-world examples that illustrate surface area and volume?
These concepts are used everywhere in daily life. Here are a few examples:
Surface Area Examples: The amount of paint required to cover a room's walls, the quantity of fabric needed to make a tent, or the area of skin on an animal which affects heat regulation.
Volume Examples: The amount of juice in a carton, the capacity of a car's fuel tank, or the amount of concrete needed to lay a foundation.
3. What is the difference between Total Surface Area (TSA) and Curved/Lateral Surface Area (CSA/LSA)?
Total Surface Area (TSA) is the sum of the areas of all surfaces of a 3D object. For a cylinder, this includes the area of the top and bottom circles plus the area of the curved side. Curved Surface Area (CSA), or Lateral Surface Area (LSA), is the area of only the curved faces, excluding the flat bases. For a cylinder, the CSA is just the area of the rectangular-like side, without the top and bottom circles.
4. How is the surface area of a combination of solids calculated?
To calculate the surface area of a combined solid (e.g., a cone on top of a cylinder), you must only add the areas of the exposed surfaces. A common mistake is to simply add the total surface areas of the individual shapes. Instead, you must identify which surfaces are covered or joined. For a cone placed on a cylinder, the base of the cone and the top of the cylinder are not exposed, so they are excluded from the final calculation.
5. How are the volumes of a cone, cylinder, and sphere related?
There is a beautiful mathematical relationship between these shapes. If a cone and a cylinder have the same base radius (r) and height (h), the volume of the cone is exactly one-third the volume of the cylinder. The volume of a sphere is related by the formula (4/3)πr³. Archimedes famously showed that the volume of a sphere is two-thirds the volume of a cylinder that perfectly encloses it.
6. Why is understanding surface area and volume important in fields like engineering and biology?
This understanding is critical for efficiency and function. In engineering, it is used to design engine parts, calculate material costs for construction, and optimise heat transfer in radiators (which have a large surface area for a small volume). In biology, the surface area to volume ratio governs how efficiently a cell can exchange nutrients and waste, and why large animals have adaptations like folded internal surfaces for digestion and respiration.
7. If an object is melted and recast into a new shape, what happens to its volume and surface area?
When an object is melted and recast, the amount of material remains the same. Therefore, its volume remains constant. However, its surface area will almost always change depending on the new shape. For example, if a solid sphere is recast into a long, thin wire, the volume of metal is the same, but the surface area of the wire will be significantly larger than that of the original sphere.
8. What is a frustum and why are its calculations considered difficult?
A frustum is the portion of a solid, typically a cone or pyramid, that remains after its top section is cut off by a plane parallel to its base. It looks like a bucket or a lampshade. Calculations for a frustum are considered difficult because they involve two different radii (one for the top base and one for the bottom base) and require using specific formulas derived from the properties of the original, complete shape. Students must carefully track both radii and the slant height to avoid errors.
9. Can an object's surface area value be numerically larger than its volume value?
Yes, it is entirely possible. The relationship between the numerical values of surface area and volume depends on the object's shape and the units of measurement. For example, a very large, thin sheet of aluminium foil has a massive surface area but a very small volume. Similarly, a cube with a side length of 0.1 cm has a surface area of 6 * (0.1)² = 0.06 cm² and a volume of (0.1)³ = 0.001 cm³, making its surface area value 60 times larger than its volume value.
