12. Surface Areas and Volumes

4 topics55 min

Chapter 12 of NCERT Mathematics for Class 10: Surface Areas and Volumes. Two sections — Outer surface of a joined solid and Capacity of a joined solid. Four videos, 55 min in all.

Worked answers to the 17 exercise questions in this chapter: Exercise 12.1 · Exercise 12.2

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Part 1

Outer surface of a joined solid

13 minDecomposing an everyday object into the basic solidsThere is no formula for a spinning top, and there never will be - tops are things people make, not shapes anyone chose to name. So the height of its cone is not a number you are ever told. It is the only height at which the two pieces will meet at all, and the explanation goes and finds it.13 min · 14 parts14 minWhich faces vanish at the join, and why you cannot simply addPaint a cone. Paint a dome. Add the two amounts up, then put the dome on the cone and paint that instead - and the number goes down. Two circles of paint, bought and never used. Working out exactly which paint gets wasted is the whole of this topic, and it is not always two circles.14 min · 14 parts

Part 2

Capacity of a joined solid

15 minWhy volumes do add even though surface areas do notPush two solids together and their surface areas do NOT add - the two faces that meet stop being outside. So when you push them together and add up the space they occupy, surely something goes wrong there too? It does not. Volumes add exactly. One sentence about what a face is settles both, and it is the same sentence.15 min · 13 parts13 minWhen a piece has been scooped out: apparent capacity against actualA juice seller's glass is 5 centimetres across and 10 tall, so it holds 196.25 cubic centimetres. Except that the bottom rises into it as a dome of solid glass, and glass is not juice. The customer loses exactly one sixth - and would lose exactly one sixth whether you took pi as 3.14, as 22 sevenths, or exactly.13 min · 13 parts

What you will be able to do

  • Look at a drawn or described object and name the basic solids it is assembled from, and how many of each
  • Explain why a joined solid's total surface area falls short of what the two pieces' total surface areas would give
  • State the additivity rule for the volume of a joined solid and give the reason, not just the rule
  • Compute the apparent capacity of a vessel from its outline alone

What this chapter assumes you already know

  • The names cuboid, cone, cylinder, sphere, hemisphere, and the fact that each is determined by very few numbers — an edge triple, a radius and height, a radius alone
  • The Class IX surface-area results for those solids: the curved surface of a cylinder, the curved surface of a cone in terms of its slant height, the surface of a sphere, and the six faces of a cuboid
  • That a cone's slant height is recovered from its radius and its vertical height by Pythagoras, and is not one of the numbers a problem usually gives you
  • Radius as half the diameter, and the habit of checking which of the two a figure has actually labelled
  • Working fluently in centimetres and metres within one problem, and converting between them before combining
  • The Class IX formulas for the curved surface of a cylinder and of a cone, and for the surface of a sphere, together with the area of a circle and of a square
  • That a hemisphere has two pieces of surface, a curved dome of area 2πr² and a flat circular face of area πr²
  • Confidence that the area of a plane region can be added and subtracted like any other measured quantity

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "There must be a formula for a top / a capsule / a rocket."
  • "The number I was given is the number the formula wants."
  • "Slant height is one of the given measurements."
  • "The two pieces meeting at a join always share a radius."
  • "A hemisphere is a basic solid, so it will be in the opening figure."
  • "Scooping and sticking are different problems."
  • "Mixed units will come out in the wash."
  • "Total surface area of a composite = TSA of one piece + TSA of the other."