Surface Areas And Volumes Formulas Definition and Solved Examples
The concept of surface areas and volumes plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing how to calculate the surface area and volume of 3D shapes is essential for solving many board exam questions. On this page, you’ll find definitions, formula tables, visual guides, and step-by-step examples to help you master “surface areas and volumes” for class 9, class 10, and beyond.
What Is Surface Area and Volume?
A surface area is defined as the total area covering the outside of a 3D object—like a box, sphere, or cylinder. Volume is the measure of the space inside that object. You’ll find this concept applied in areas such as solid geometry, mensuration, and practical activities like packaging or storage.
Surface Area and Volume Formula Table
Here’s a quick-reference table of key surface area and volume formulas for commonly tested shapes:
| Shape | Total Surface Area (TSA) | Curved/ Lateral Surface Area (CSA/LSA) | Volume |
|---|---|---|---|
| Cube (side = a) | 6a² | 4a² | a³ |
| Cuboid (l, b, h) | 2(lb + bh + hl) | 2h(l + b) | l × b × h |
| Right Circular Cylinder (r, h) | 2πr(r + h) | 2πrh | πr²h |
| Right Circular Cone (r, l, h) | πr(r + l) | πrl | (1/3)πr²h |
| Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (r) | 3πr² | 2πr² | (2/3)πr³ |
Shape-wise Explanation and Diagrams
Let’s quickly break down the surface area and volume for each solid:
- Cube: All sides equal. Surface area covers all 6 faces. Detailed cube formula guide.
- Cuboid: Rectangle box, each face can have different dimensions.
- Cylinder: Has two circular bases and a curved surface. TSA adds both bases plus the curve.
- Cone: Circular base and a curved surface slanting upwards. Use slant height (l) in formulas.
- Sphere: Perfect ball. Same formula for surface area (no base).
- Hemisphere: Half of a sphere. TSA also includes the flat base area.
Visualizing 3D shapes helps – try drawing nets of solids or use interactive tools in Vedantu’s solid shapes section for better clarity!
Word Problem – Stepwise Solution
Example: Find the volume and the total surface area of a cylinder with base radius 4 cm and height 10 cm.
1. Write the known values: r = 4 cm, h = 10 cm2. TSA = 2πr(r+h) = 2 × 3.14 × 4 × (4 + 10) = 2 × 3.14 × 4 × 14 = 351.68 cm²
3. Volume = πr²h = 3.14 × 16 × 10 = 502.4 cm³
4. Final answers: TSA is 351.68 cm² and volume is 502.4 cm³
Difference Between TSA, CSA, and LSA
| Abbreviation | Full Form | What’s Included? |
|---|---|---|
| TSA | Total Surface Area | Curved + Base(s) + Top (all faces) |
| CSA | Curved Surface Area | Only the curved/lateral surface (no bases) |
| LSA | Lateral Surface Area | Same as CSA (alternate name) |
CBSE & NCERT Exam Tips
- Memorize formulas using a cheat sheet for cylinders, cones, spheres, and combinations.
- Watch for curved vs. total areas in board questions—marks get lost over this!
- Use Vedantu’s complete class 10 mensuration formulas page to revise before exams.
- Word problems may combine 2–3 shapes—draw, label, and apply formulas step-by-step.
Speed Tricks and Quick Shortcuts
For fast calculation, many students remember π as 22/7, and use tables for squares and cubes. Some use the “net” method for combinations: Add up surface areas of individual solids, subtract joined faces.
- For spheres and hemispheres, surface area and volume are always related by multiplying/dividing by 2 (for hemisphere).
- Curved surface area of cylinder: Just the rectangle with width = height and length = circumference of base.
Learn more such mental shortcuts in Vedantu’s live doubt-clearing classes or refer to our mensuration shortcuts article.
Try These Yourself
- Find the TSA and volume of a cube with side 6 cm.
- A cuboid has dimensions 8 cm × 4 cm × 2 cm. Calculate its surface area and volume.
- What is the difference between CSA and TSA of a cylinder of radius 5 cm and height 10 cm?
- If the volume of a sphere is 113.04 cm³, find its radius.
Relation to Other Concepts
The idea of surface areas and volumes connects closely with 2D area and perimeter, area of triangles, and 2D plane figure calculations. It also forms a foundation for advanced geometry and practical science experiments—measuring storage, liquids, and material cost.
Classroom & Revision Tip
A simple way to remember TSA and CSA: TSA is “Touching all sides,” CSA is “Curved Only!” Use color-coding for nets and formula flashcards. Vedantu’s teachers suggest regular 5-minute formula drills to avoid silly mistakes before exams.
We explored surface areas and volumes—from definitions and formulas to word problems, differences between TSA/CSA, and their applications in exams. Continue practicing with Vedantu for more real-life problems, interactive diagrams, and instant calculator tools.
Must-Visit Links for Deep Dives:
- Surface Area of Cube
- Volume of Cuboid
- Mensuration Formulas Class 10
- Surface Area and Volume Calculator
FAQs on Surface Areas And Volumes Complete Guide with Formulas and Applications
1. What is surface area in Maths?
Surface area is the total area of all the outer faces of a three-dimensional (3D) object. In Surface Areas and Volumes, it tells us how much material is needed to cover the outside of a solid.
- Measured in square units (cm², m², etc.).
- Includes curved surfaces for shapes like cylinders and spheres.
- Example: A cube with side 4 cm has surface area = 6a² = 6 × 4² = 96 cm².
2. What is volume in geometry?
Volume is the amount of space occupied by a three-dimensional solid. In mensuration, it shows how much a container can hold.
- Measured in cubic units (cm³, m³, etc.).
- For a cube of side a, volume = a³.
- Example: If side = 5 cm, volume = 5³ = 125 cm³.
3. What is the formula for the surface area of a cube?
The formula for the surface area of a cube is 6a², where a is the side length. A cube has 6 equal square faces.
- Area of one face = a²
- Total surface area = 6 × a²
- Example: If a = 3 cm, surface area = 6 × 9 = 54 cm².
4. What is the formula for the volume of a cuboid?
The volume of a cuboid is l × b × h, where l = length, b = breadth, and h = height. This formula multiplies the three dimensions of the solid.
- All dimensions must be in the same unit.
- Example: If l = 6 cm, b = 4 cm, h = 3 cm, volume = 6 × 4 × 3 = 72 cm³.
5. What is the curved surface area of a cylinder?
The curved surface area (CSA) of a cylinder is 2πrh, where r is the radius and h is the height. It represents only the side surface, not the circular bases.
- Total surface area = 2πr(h + r)
- Example: If r = 7 cm and h = 10 cm, CSA = 2 × π × 7 × 10 = 140π cm².
6. What is the formula for the volume of a cylinder?
The volume of a cylinder is πr²h, where r is the radius of the base and h is the height. It is the area of the circular base multiplied by height.
- Base area = πr²
- Volume = πr²h
- Example: If r = 3 cm and h = 5 cm, volume = π × 9 × 5 = 45π cm³.
7. What is the surface area of a sphere?
The surface area of a sphere is 4πr², where r is the radius. A sphere has no edges or vertices, only a curved surface.
- Measured in square units.
- Example: If r = 7 cm, surface area = 4 × π × 49 = 196π cm².
8. What is the volume of a sphere?
The volume of a sphere is (4/3)πr³, where r is the radius. This formula calculates the space inside the sphere.
- Measured in cubic units.
- Example: If r = 3 cm, volume = (4/3) × π × 27 = 36π cm³.
9. What is the difference between surface area and volume?
Surface area measures the outer covering of a solid, while volume measures the space inside the solid. These are two key concepts in Surface Areas and Volumes.
- Surface area → square units (cm², m²)
- Volume → cubic units (cm³, m³)
- Example: A box can have large surface area but smaller volume depending on dimensions.
10. How do you solve word problems on surface areas and volumes?
To solve word problems on surface areas and volumes, first identify the solid and required measure, then apply the correct formula with consistent units. Follow these steps:
- Read the question carefully and note given dimensions.
- Convert all measurements to the same unit.
- Select the correct formula (e.g., 6a², πr²h, 4πr²).
- Substitute values and calculate accurately.
- Write the final answer with correct square or cubic units.
