Surface Areas & Volumes
CBSE Class 10 – Mathematics
(All units in cm unless otherwise specified)
Why do we study this?
Surface area tells how much material is needed to cover a solid, while volume tells how much space it occupies. These concepts are fundamental in engineering, architecture, physics, and everyday problem‑solving.
1. Basic Ideas
| Term | Meaning |
|---|---|
| Surface Area (SA) | Total area of all faces of a solid. |
| Volume (V) | Amount of space a solid occupies. |
| Units | SA → cm² (square centimetres) V → cm³ (cubic centimetres) |
Tip: When you see “area” → square units, “volume” → cubic units.
2. Surface Area & Volume of Common Solids
All formulas are derived from the definition of the solid.
Use the appropriate formula based on the shape.
2.1 Cube
| Formula | Description |
|---|---|
SA = 6a² |
6 faces, each area a². |
V = a³ |
Side³ gives volume. |
2.2 Cuboid (Rectangular Prism)
| Formula | Description |
|---|---|
SA = 2(ab + bc + ca) |
Sum of areas of all 6 faces. |
V = abc |
Product of length, breadth, height. |
2.3 Sphere
| Formula | Description |
|---|---|
SA = 4πr² |
Surface of a round ball. |
V = 4/3 πr³ |
Volume inside the sphere. |
2.4 Cylinder
| Formula | Description |
|---|---|
SA = 2πr(r + h) |
Lateral area 2πrh + top & bottom 2πr². |
V = πr²h |
Base area × height. |
2.5 Right Circular Cone
| Formula | Description |
|---|---|
SA = πr(r + l) |
Lateral area πrl + base πr². |
V = 1/3 πr²h |
Cone’s volume. |
where l = √(r² + h²) |
Slant height (from Pythagoras). |
2.6 Pyramid (any base)
| Formula | Description |
|---|---|
SA = Area(base) + 1/2 * Perimeter(base) * SlantHeight |
Base + lateral faces. |
V = 1/3 * Area(base) * Height |
Height is perpendicular to base. |
2.7 Prism (any base)
| Formula | Description |
|---|---|
SA = 2 * Area(base) + Perimeter(base) * Height |
Bases + lateral faces. |
V = Area(base) * Height |
Base area × height. |
Special Note: A triangular prism has a triangular base; a rectangular prism is a cuboid.
3. Key Points to Remember
| # | Concept | Quick Check |
|---|---|---|
| 1 | Identify the solid first (cube, cuboid, sphere, etc.). | |
| 2 | Differentiate between lateral and total surface area. | |
| 3 | Use the right formula—remember the constants (π, 1/3, 2, 6). |
|
| 4 | Dimensions must be in the same units before plugging into formulas. | |
| 5 | Check your final units: SA in cm², V in cm³. | |
| 6 | Area of a circle = πr². It is used in many surface area formulas. |
|
| 7 | Slant height (l) = √(r² + h²) for cones, pyramids. |
|
| 8 | For prisms/pyramids: keep base area and height separate; they are not the same as the slant height. |
4. Worked Example
Problem:
*A right circular cone has a radius of