7. Area

Part II, printed Chapter 79 topics2 h

Chapter 7 of NCERT Mathematics for Class 8 (Part II, printed Chapter 7): Area. Three sections — What area measures, Triangles, and every polygon after them and The special quadrilaterals. Nine videos, 2 h in all.

Worked answers to the 36 exercise questions in this chapter →

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Part 1

What area measures

10 minArea as a count of unit squaresOne square cut into four equal parts. The same square cut so the boundaries step in and out, and no piece looks like any other.10 min · 12 parts9 minWhy perimeter cannot stand in for areaEvery flat shape gives you two numbers: how far round the edge, and how much is inside. Neither one decides the other.9 min · 12 parts

Part 2

Triangles, and every polygon after them

10 minWhy the area of a triangle is half base times heightDraw a triangle inside a rectangle and slide its top corner along the top edge. The shape changes at every step and the area never does.10 min · 12 parts9 minTriangles with the same base and height have the same areaBoth diagonals of a rectangle cut it into four triangles that plainly do not match — two wide and low, two narrow and tall.9 min · 12 parts11 minAny polygon is a pile of trianglesA quadrilateral with no name: four different sides, no two parallel, no formula anywhere. One diagonal solves it.11 min · 12 parts

Part 3

The special quadrilaterals

10 minArea of a parallelogram, by turning it into a rectangleThe plan is not to find a formula. It is to find a rectangle.10 min · 12 parts10 minArea of a rhombus from its diagonalsA rhombus is a parallelogram, so base times height already covers it. Half the product of the diagonals is a second route — and it needs a condition.10 min · 12 parts11 minArea of a trapezium, derived two different waysHalf the height times the sum of the parallel sides is not a fourth thing to memorise, and the half is not something to remember.11 min · 12 parts10 minUnits for real areas, and why converting between them squares the length factorOne square inch is 6.4516 square centimetres. An inch is only 2.54 centimetres — and the gap between those two facts is the whole topic.10 min · 12 parts

What you will be able to do

  • State what quantity is being counted when an area is reported, and name the square that is doing the counting
  • State what a measure of area is required to do that boundary length fails to do
  • Show that a triangle on one side of a rectangle, with its apex anywhere on the opposite side, is half that rectangle
  • State the condition under which two triangles must have equal areas, and distinguish it from congruence
  • State the two facts that together make every polygon computable, and say which of them the chapter proves and which it asserts
  • Construct a height of a given parallelogram and say why the article is "a" rather than "the"
  • State why a rhombus needs no separate area formula, and why it is given one anyway
  • Cut a trapezium into a rectangle and two right triangles, and justify that the middle piece really is a rectangle
  • Compute the area of a stated rectangular object from its two sidelengths, with the unit

What this chapter assumes you already know

  • Multiplication of two whole numbers, and division as the inverse of it
  • What a square and a rectangle are, and that a rectangle's opposite sides are equal
  • Perimeter as the total length around a closed figure
  • That two figures are congruent when one can be laid exactly on the other
  • Reading a measurement off a labelled diagram, including a label written inside a shape rather than beside it
  • Perimeter of a rectangle, and that it depends on both sidelengths
  • Multiplication and addition of one- and two-digit numbers, and of simple decimals
  • That a statement about "all" shapes is destroyed by one counter-instance

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.

  • "Area is a formula you apply."
  • "Cutting a shape up changes its area."
  • "Rearranging can create area — look, the square got bigger."
  • "cm² means 'centimetres, and then square the number'."
  • "You can only measure whole unit squares, so awkward shapes have no area."
  • "Half the rectangle needs half the unit squares to come out whole."
  • "Equal perimeter means equal area."
  • "A missing sidelength needs a ruler."