CLASS 8 MATHS · PART 2 · CHAPTER 7 · STUDY NOTES

Area

Understand area through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.

UNDERSTAND · APPLY · CHECK

Area counts unit squares; perimeter counts the boundary

For a rectangle, rows × squares per row gives length × width.

Same perimeter; different covered area

y (units)

x (units)

Highlight a line or construction

6 by 4 · 6 by 4
7 by 3 · Same perimeter 20, area 21.

1Area 24 · (3, 3.5)
2Area 21 · (6.5, 1.5)

Equal scales and common corner allow exact comparison. The outlines overlap intentionally; their areas are evaluated separately. Equal units on both axes.

Read the plotted values

6 by 4

x	y
0	0
6	0
6	4
0	4
0	0

7 by 3

x	y
0	0
7	0
7	3
0	3
0	0

Area: The amount of surface covered by a region, measured in square units.
Perimeter: The total length around a boundary, measured in length units.

A 6 by 4 rectangle has area 24 square units and perimeter 20 units. A 7 by 3 rectangle also has perimeter 20 but area 21.

Equal perimeter does not imply equal area. If every length doubles, every area becomes four times as large.

Take a moment. What do you think?

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Question

Find area and perimeter of a 9 cm by 2 cm rectangle.

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NCERT reference: chapter PDF pages 1, 2, 3.

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Subtract a hole; count an overlap only once

Break composite shapes into known regions without leaving gaps or double-counting.

The crossing is counted twice

y (units)

x (units)

Highlight a line or construction

Horizontal strip · Horizontal strip
Vertical strip · Vertical strip
Shared crossing · Shared crossing

1Subtract overlap 1 · (3.5, 2.5)

The union of the two strips covers 13 square units. The third shaded region identifies their 1-square-unit overlap; it is not extra area. Equal units on both axes.

Read the plotted values

Horizontal strip

x	y
0	2
8	2
8	3
0	3
0	2

Vertical strip

x	y
3	0
4	0
4	6
3	6
3	0

Shared crossing

x	y
3	2
4	2
4	3
3	3
3	2

Composite area: An area found by adding or subtracting simpler component regions.

An 8 by 6 outer rectangle surrounding a 6 by 4 inner rectangle leaves a path area 48 − 24 = 24. Moving the inner rectangle within it does not change that difference.

Two perpendicular strips of width 1 across an 8 by 6 rectangle have area 8×1 + 6×1 − 1×1 = 13. Their crossing was included twice, so subtract it once.

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Question

An outer 10 by 7 rectangle contains a 6 by 3 hole. What area remains?

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NCERT reference: chapter PDF pages 4, 5.

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Use the perpendicular height, not a sloping side

A triangle covers half a rectangle or parallelogram with the same base and height.

An altitude can lie outside

y (units)

x (units)

Highlight a line or construction

Triangle · Triangle
Extended base · The altitude reaches this extension.
Perpendicular height · Height 4 from vertex to base line.

1Base 6 · (3, 0)
2Height 4 · (-2, 2)

Exact obtuse triangle with base 6 and perpendicular height 4. The slanted side is not used as the height. Equal units on both axes.

Read the plotted values

Triangle

x	y
0	0
6	0
-2	4
0	0

Extended base

x	y
-2	0
0	0

Perpendicular height

x	y
-2	0
-2	4

Altitude: The perpendicular segment from a vertex to the line containing the opposite side.

Area = ½ × base × perpendicular height. If base is 6 and height 4, area is 12 square units.

For an obtuse triangle, the altitude may land outside the base segment. The formula still works because height measures distance to the base’s line.

1
Choose a base and its corresponding vertex.
2
Drop a perpendicular to the base line, extending it if necessary.
3
Multiply base by height.
4
Halve the product and write square units.

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Question

A triangle has area 30 cm² and base 12 cm. Find its perpendicular height.

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NCERT reference: chapter PDF pages 3, 6, 7, 8, 11.

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The same base and height give equal areas

Sliding the top vertex along a parallel line changes shape but keeps area.

Move the vertex; keep the height

y (units)

x (units)

Highlight a line or construction

Triangle A · Apex (1,4).
Triangle B · Apex (5,4).
Parallel line · Every apex here has height 4.

1Common base 6 · (3, 0)
2Height stays 4 · (3, 4)

Both exact triangles have area 12, regardless of their different outlines. Equal units on both axes.

Read the plotted values

Triangle A

x	y
0	0
6	0
1	4
0	0

Triangle B

x	y
0	0
6	0
5	4
0	0

Parallel line

x	y
-1	4
7	4

Equal-area triangles: Triangles whose areas match, even when their shapes or side lengths differ.

Any triangle on a base of length 6 with its third vertex on a parallel line 4 units away has area 12.

A median joins a vertex to the midpoint of the opposite side. Its two triangles have equal bases and a common height, so their areas are equal.

Let’s look a little closer.

Go deeper: Area and shortest perimeter answer different questions

On a fixed base between parallel lines, area stays constant. Perimeter can change. With the apex unrestricted on the other parallel line, the minimum occurs above the base midpoint; reflection turns the two sloping sides into a straight shortest route.

Joining two side midpoints gives a smaller similar triangle with half the side scale, so its area is one quarter, not one half.

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Question

Does equal area prove two triangles are congruent?

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NCERT reference: chapter PDF pages 8, 9, 10, 11, 12.

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Split a polygon into triangles with known heights

Choose diagonals inside the polygon, then add the triangle areas.

One diagonal; two heights

y (units)

x (units)

Highlight a line or construction

Upper triangle · Upper triangle
Lower triangle · Lower triangle
Upper height · Perpendicular distance 3.
Lower height · Perpendicular distance 2.

1Diagonal 8 · (4, 0)
23 · (3, 2)
32 · (5, -1)

Exact convex quadrilateral partition. The two triangles share diagonal 8 and have no overlapping interiors. Equal units on both axes.

Read the plotted values

Upper triangle

x	y
0	0
8	0
3	3
0	0

Lower triangle

x	y
0	0
8	0
5	-2
0	0

Upper height

x	y
3	0
3	3

Lower height

x	y
5	0
5	-2

Triangulation: Dividing a polygon into non-overlapping triangles using diagonals.

A quadrilateral with diagonal 8 and perpendicular heights 3 and 2 on its opposite sides has area ½×8×3 + ½×8×2 = 20.

The diagonal is the base for both triangles. A general polygon needs enough measurements for all its triangle areas; a drawing alone does not supply missing lengths.

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Question

A quadrilateral’s diagonal is 10; its two perpendicular heights are 4 and 5. Find its area.

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NCERT reference: chapter PDF pages 12, 13.

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Move a triangle to make a rectangle

A dissection changes the outline while preserving all the pieces.

Same pieces: parallelogram becomes rectangle

Orange triangle: Moves without changing shape or area.
Blue piece: Stays in place.

Exact equal-scale dissection: the orange right triangle is translated five length units to the right. The resulting rectangle has the same two pieces.

Dissection: Cutting a region into pieces and rearranging them without changing the total area.

A parallelogram’s area is base × perpendicular height. Its slanting side is generally longer than the height.

For base 5 and height 3, area is 15. The cut triangle can be translated to the other end to create a 5 by 3 rectangle.

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Question

A parallelogram has area 42 cm² and base 7 cm. Find its height.

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NCERT reference: chapter PDF pages 14, 15, 16, 17.

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The diagonals give four right triangles

A rhombus’s diagonals bisect each other at right angles.

Four triangles fill the rhombus

y (units)

x (units)

Highlight a line or construction

Rhombus · Rhombus
Diagonal 8 · Horizontal diagonal.
Diagonal 6 · Vertical diagonal.

1Half 4 · (2, 0)
2Half 3 · (0, 1.5)

Exact rhombus: equal sides 5, perpendicular diagonals 8 and 6. Each of four right triangles has area 6. Equal units on both axes.

Read the plotted values

Rhombus

x	y
0	3
4	0
0	-3
-4	0
0	3

Diagonal 8

x	y
-4	0
4	0

Diagonal 6

x	y
0	-3
0	3

Rhombus area: Half the product of its two diagonal lengths: A = d₁d₂/2.

For diagonals 8 and 6, the four right triangles each have legs 4 and 3. Total area is 4×½×4×3 = 24.

This is the same as ½×8×6. The base × perpendicular height formula also works because every rhombus is a parallelogram.

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Question

Find a rhombus’s area with diagonals 12 cm and 5 cm.

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NCERT reference: chapter PDF pages 17, 18, 19, 22.

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Two copies make a parallelogram

Add the parallel side lengths, multiply by height, then halve.

Two equal copies fill a parallelogram

y (units)

x (units)

Highlight a line or construction

First trapezium · First trapezium
Rotated copy · Rotated copy
Height · Perpendicular height 3.

1Combined base 9 · (4.5, 0)
2Height 3 · (1, 1.5)

Exact coordinate construction: each trapezium has parallel sides 6 and 3, height 3 and area 13.5. Together they have area 27. Equal units on both axes.

Read the plotted values

First trapezium

x	y
0	0
6	0
4	3
1	3
0	0

Rotated copy

x	y
6	0
9	0
10	3
4	3
6	0

Height

x	y
1	0
1	3

Trapezium area: For parallel sides a,b and perpendicular height h, A = ½(a + b)h.

For parallel sides 6 and 3 with height 3, the area is ½×9×3 = 13.5.

Rotate a matching copy and attach it along a nonparallel side. The combined parallelogram has base a + b and the same height h, explaining why one copy takes half its area.

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Question

Find the area for parallel sides 8 m and 12 m and height 5 m.

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NCERT reference: chapter PDF pages 19, 20, 21, 22, 23.

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Square the conversion factor; count whole tiles

Area units change in two length directions.

Tile dimensions fit the floor exactly

y (units)

x (units)

Highlight a line or construction

Floor · 3 by 2 metres.
One tile · Side 0.5 m and area 0.25 m².
Column boundary marks · Six columns of equal width.
Row boundary marks · Four rows of equal height.

10.5 × 0.5 · (0.25, 0.25)
2Floor width 3 m · (1.5, 2)

Exact aligned rectangle: six half-metre columns and four half-metre rows give 24 whole tiles. Boundary markers identify equal intervals; only one tile is shaded. Equal units on both axes.

Read the plotted values

Floor

x	y
0	0
3	0
3	2
0	2
0	0

One tile

x	y
0	0
0.5	0
0.5	0.5
0	0.5
0	0

Column boundary marks

x	y
0.5	0
1	0
1.5	0
2	0
2.5	0

Row boundary marks

x	y
0	0.5
0	1
0	1.5

Square-unit conversion: A conversion using the square of the corresponding length factor.

Unit	Exact equivalent
1 m²	10,000 cm²
1 km²	1,000,000 m²
1 in²	6.4516 cm²
1 ft²	144 in²
1 acre	43,560 ft²

1 inch = 2.54 cm exactly, so 1 in² = 6.4516 cm². 1 ft = 12 in gives 1 ft² = 144 in².

A 3 m by 2 m floor has area 6 m². Square tiles of side 0.5 m have area 0.25 m², so an exact aligned layout uses 6/0.25 = 24 tiles. A real layout may need extra tiles for cutting and wastage.

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Question

How many cm² are in 0.4 m²?

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NCERT reference: chapter PDF pages 23, 24.

Area Class 8 Notes

Chapter notes

Area

Area counts unit squares; perimeter counts the boundary

Same perimeter; different covered area

Subtract a hole; count an overlap only once

The crossing is counted twice

Use the perpendicular height, not a sloping side

An altitude can lie outside

The same base and height give equal areas

Move the vertex; keep the height

Split a polygon into triangles with known heights

One diagonal; two heights

Move a triangle to make a rectangle

Same pieces: parallelogram becomes rectangle

The diagonals give four right triangles

Four triangles fill the rhombus

Two copies make a parallelogram

Two equal copies fill a parallelogram

Square the conversion factor; count whole tiles

Tile dimensions fit the floor exactly

Now practise Area, free