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CBSE Class 10 · Mathematics

Surface Areas and Volumes

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Surface Areas and Volumes” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Surface Area of a Combination of Solids

A combination’s surface is the outer skin that remains after you join solids — subtract the hidden faces that got glued. A hemisphere stuck on a cube’s face: drop that square (or that disc) from the sum, as the join taught. Surface is the leftover skin, not the sum of every solid as if they sat apart.

Adding a full cube and a full sphere when they share a face double-counts the join.

Figure. Side elevation of a cone on a cylinder with a shared radius. TSA is the exposed skin: curved cylinder, curved cone, and the open base. The circular join is glued inside and is not added.

How it works

  1. Name the solids and the glued faceThe join.
  2. Add the skins, then drop the hidden partsThe outer.
  3. Keep cm²Surface.

2Volume of a Combination of Solids

A combination’s volume adds the insides: the hidden face does not remove litres. A cube of side 6 cm plus a hemisphere of radius 3 cm on a face is 216 + (2/3)π×27, with π named. Volume is the fill, not the skin.

Subtracting the join from the volume as if glue ate the water is a miss.

Figure. Volume adds the fills. Cylinder πr²H plus cone (1/3)πr²h. The join is not subtracted unless a piece was hollowed out.

How it works

  1. Name each solid’s fillCube, hemisphere…
  2. Add themThe combination volume.
  3. Keep cm³Volume.

6 cm cube plus r = 3 cm hemisphere

A hemisphere of radius 3 cm sits on one face of a 6 cm cube (join is a disc). Find the volume, π = 22/7.

  • Cube6³ = 216 cm³
  • Hemisphere(2/3)πr³ = (2/3)×(22/7)×27 = 396/7 cm³
  • Sum216 + 396/7 cm³

Pro tip. Volume adds; the join does not eat the fill.

3A definition is a test you can run

Surface / volume of a join is a test: name the glued face and whether you are walking the skin or filling the inside. If you only say “combination”, you have a heading.

A toy photo is not the test.

Figure. Run the motion test first. The school name is pinned on after the object rolls every way, slides on a face, or rolls only on an edge.

How it works

  1. Name the solids and the joinThe object.
  2. Say skin (drop hidden) or fill (add insides)The test.
  3. Then the word has contentThe definition ran.

4Name the given before the unknown

The given are the dimensions and what was glued. The unknown is skin or fill. Copy “hemisphere on a face” before you add a full sphere.

Using r = 6 because the cube is 6, when the hemisphere sat on a face as r = 3, is a silent given.

Figure. Write r, H, h (and l if given) before reaching for TSA or volume. The unknown is one sentence; the givens are the marks on the elevation.

How it works

  1. Copy the sides and the joinThe given.
  2. Name CSA/TSA or volumeThe unknown.
  3. Then drop hidden or add insidesGiven first.

5One worked case is enough at this class

One join is enough. A page of the same 216-plus-hemisphere clone does not add a new cone.

Ten twins of a cube-cap are still one idea.

Figure. One solid, one number set: r=6, H=10, h=8, l=10 (6-8-10). Exposed curved area is 2πrH + πrl = 180π. Do not start a second toy until this ledger closes.

How it works

  1. Run one surface or one volumeThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

An ice-cream photo is a setting. The maths is cone + hemisphere (or the taught pair) with a named join. A scoop is not r until you measure.

A brand-cone is not the formula.

Figure. A box that someone labelled “sphere” is still a cuboid: it slides on a face. The school name is a result of the test, not a sticker you apply first.

How it works

  1. Keep the snack as a storyA setting.
  2. Name the two solids and the joinThe maths.
  3. Then skin or fillDo not swap.

7Check by the opposite action or the opposite test

Check volume by adding the parts another way; check surface by listing every remaining face. A cm² that equals a cm³ writing is a unit-miss, not a check.

Comparing 216 to 396/7 without units is mute.

Figure. Opposite check: if you computed TSA, ask whether a glued face was wrongly added. If you computed volume, ask whether both fills were added and nothing was double-counted.

How it works

  1. State the skin or the fillThe claim.
  2. Rebuild the remaining faces or the partsThe check.
  3. Mend if they disagreeHonest.

8Keep the claim at this chapter, not the next

This chapter’s claim stops at combined skin and fill. It does not steal grouped-data mean, and it does not start a calculus volume.

A page that already “finds a median class” here misses statistics.

Figure. Class 10 names r, h, l and adds known pieces. Surfaces of revolution and integrals wait for a later book.

How it works

  1. Keep join-skin and join-fillThis chapter.
  2. Leave mean/mode/median of grouped data for statisticsThe next map.
  3. Do not treat the map as already answeredClaim-size.
When two solids are glued, volume
  1. Adds the insides — the join does not eat the fill
  2. Must subtract the glued face from the litres
  3. Equals the surface

Skin drops hidden faces; fill still adds.

Notes

  • The official chapter title is “Surface Areas and Volumes”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • combination surface: add skins, drop hidden faces
  • combination volume: add insides
  • hemisphere V = (2/3)πr³

Recap

Hold these pegs from the official chapter “Surface Areas and Volumes”. The wording is ExamMaster’s teaching, not a textbook recap.

Surface Area of a Combination of Solids
A combination’s surface is the outer skin that remains after you join solids — subtract the hidden faces that got glued.
Volume of a Combination of Solids
A combination’s volume adds the insides: the hidden face does not remove litres.
A definition is a test you can run
Surface / volume of a join is a test: name the glued face and whether you are walking the skin or filling the inside.
Name the given before the unknown
The given are the dimensions and what was glued.
One worked case is enough at this class
One join is enough. A page of the same 216-plus-hemisphere clone does not add a new cone.
A miss: swapping the school name for the picture
An ice-cream photo is a setting. The maths is cone + hemisphere (or the taught pair) with a named join. A scoop is not r until you measure.

Practise Surface Areas and Volumes

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