Schooling · Class 10 · Mathematics · Surface Areas and Volumes · Chapter on a page
CBSE Class 10 · Mathematics · NCERT Mathematics
Surface Areas and Volumes: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- The surface area of a combination of…
- When two solids are joined, the…
- When a solid is melted and recast…
- A frustum of a cone is the portion…
- The slant height of a frustum can be…
- Always identify which surfaces are…
- Formulas and facts
- Basic solids (for reference)
- Curved surface area of cylinder = 2πrh
- Total surface area of cylinder =…
- Volume of cylinder = πr²h
- Curved surface area of cone = πrl…
- Volume of cone = (1/3)πr²h
- Surface area of sphere = 4πr²
- Watch out for
- Adding total surface areas of each…
- Using the wrong formula for slant…
- Forgetting to convert units when…
- Using curved surface area formula…
- Assuming the number of objects must…
What this chapter is about
This chapter extends your knowledge of finding surface areas and volumes of basic solids (cuboid, cylinder, cone, sphere) to more complex situations. You will learn to calculate these measurements for combinations of solids — objects formed by joining two or more basic shapes together, such as a toy shaped like a cone attached to a hemisphere, or a storage tank made of a cylinder with hemispherical ends.
Key ideas
- The surface area of a combination of solids equals the sum of the curved/lateral surface areas of each part, minus any surfaces that are hidden or joined internally.
- When two solids are joined, the total volume equals the sum of their individual volumes.
- When a solid is melted and recast into another shape, the volume remains unchanged — this is the principle of conservation of volume.
- A frustum of a cone is the portion that remains after cutting a cone by a plane parallel to its base and removing the smaller cone formed above.
- The slant height of a frustum can be found using the Pythagorean theorem: l = √[h² + (R − r)²], where h is the vertical height, R is the larger radius, and r is the smaller radius.
- Always identify which surfaces are exposed (painted, covered, or visible) before calculating surface area.
Formulas and facts to remember
- Basic solids (for reference):
- Curved surface area of cylinder = 2πrh
- Total surface area of cylinder = 2πr(r + h)
- Volume of cylinder = πr²h
- Curved surface area of cone = πrl (where l is slant height)
- Volume of cone = (1/3)πr²h
- Surface area of sphere = 4πr²
- Volume of sphere = (4/3)πr³
- Curved surface area of hemisphere = 2πr²
- Volume of hemisphere = (2/3)πr³
- Frustum of a cone:
- Volume of frustum = (1/3)πh(R² + r² + Rr), where R and r are the radii of the two circular ends and h is the height.
- Curved surface area of frustum = π(R + r)l, where l = √[h² + (R − r)²] is the slant height.
- Total surface area of frustum = π(R + r)l + πR² + πr² (includes both circular ends).
- Conversion principle:
- Volume before reshaping = Volume after reshaping.
Common mistakes
- Adding total surface areas of each solid instead of only the exposed surfaces → Identify and exclude the joined or hidden surfaces before adding.
- Using the wrong formula for slant height of frustum; writing l = √(h² + R² + r²) → The correct formula is l = √[h² + (R − r)²].
- Forgetting to convert units when capacity is asked in litres → Remember 1 litre = 1000 cm³; always check the unit asked.
- Using curved surface area formula when total surface area is needed → Read carefully whether the problem asks for curved, lateral, or total surface area.
- Assuming the number of objects must be a whole number and rounding up → When recasting, round down because you cannot make a fraction of an object from leftover material.
Quick revision
- For combined solids: add volumes; add only exposed surface areas.
- Melting and recasting problems: equate volumes before and after.
- Frustum volume: (1/3)πh(R² + r² + Rr); slant height: l = √[h² + (R − r)²].
- Always draw a rough figure and label dimensions before calculating.
- Convert cm³ to litres by dividing by 1000.