Class 10 / Maths
Class 10 Maths Surface Areas and Volumes Notes
Notes on surface area and volume of cubes, cuboids, cylinders, cones, spheres, and combinations of solids.
Topic introduction
Surface Areas and Volumes questions become easier when you first identify the solid and the exact quantity asked: curved surface area, total surface area, or volume. For combined solids, external surfaces and hidden surfaces must be handled carefully.
Key points
- Curved surface area, total surface area, and volume are different quantities.
- A cylinder has CSA = 2 pi rh and volume = pi r^2 h.
- A cone has CSA = pi rl and volume = 1/3 pi r^2 h.
- For combined solids, split the figure into familiar solids and add or subtract surface areas carefully.
Definitions
Curved surface area: Area of only the curved surface of a solid.
Total surface area: Area of all exposed surfaces of a solid.
Volume: Space occupied by a three-dimensional solid.
Slant height: The distance from the vertex of a cone to a point on the circular edge.
Formulas
- Cuboid volume = lbh
- Cube TSA = 6a^2
- Cylinder CSA = 2 pi rh, volume = pi r^2h
- Cone CSA = pi rl, volume = 1/3 pi r^2h
- Sphere surface area = 4 pi r^2, volume = 4/3 pi r^3
Examples
- Cylinder with r = 7 cm and h = 15 cm has CSA = 2 x pi x 7 x 15.
- A solid melted and recast keeps the same volume.
- When a cone is mounted on a cylinder, the common circular face is hidden.
Formula discipline
Write the solid name, given values, required quantity, and formula before substituting numbers.
Check whether the question asks for curved surface area, total surface area, or volume.
Combinations and conversions
For combinations, hidden faces are usually not included in external surface area.
For conversion of solids, volume remains the same unless material is lost.
Common mistakes
- Using TSA when the question asks for CSA.
- Including hidden faces in external surface area of combined solids.
- Forgetting to cube radius in sphere volume.
- Using slant height instead of vertical height in cone volume.
- Not converting units before substituting values.
Quick practice
- Find the volume of a cylinder with radius 7 cm and height 10 cm.
- Find the CSA of a cone with radius 6 cm and slant height 10 cm.
- Find the TSA of a cube of side 5 cm.
- A sphere has radius 3 cm. Find its volume.
Related practice
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