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Class 9 · Maths · Chapter 6 · NCERT Class 9 Maths

Measuring Space: Perimeter and Area Class 9 Notes

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Chapter notes

CLASS 9 MATHS · CHAPTER 6 · STUDY NOTES

Measuring Space: Perimeter and Area

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

01

UNDERSTAND · APPLY · CHECK

Perimeter follows the boundary

Distinguish a boundary length from a covered area.

Around is different from inside

y (units)

x (units)

  • Boundary · The sides define the region.
  1. 16 cm · (3, 0)
  2. 23 cm · (6, 1.5)

Boundary length 18 cm; enclosed area 18 cm². One graph unit represents 1 cm. Equal units on both axes.

Read the plotted values
Boundary
xy
00
60
63
03
00
Perimeter
Total distance around a closed shape.
Area
The amount of plane region covered, measured in square units.

For a 6 cm by 3 cm rectangle, perimeter = 2(6 + 3) = 18 cm. Its area = 6 × 3 = 18 cm². The equal numbers do not make the quantities the same.

Internal dividing lines are not part of the outer perimeter. If the shape has a hole, say explicitly whether the question includes the hole’s boundary.

Take a moment. What do you think?

PAUSE & TRY · think first, then check with a free account

Question

What units should we use for a garden’s fence length and grass-covered area?

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

02

UNDERSTAND · APPLY · CHECK

Every circle has the same C/d ratio

Connect diameter, radius and circumference.

Diameter spans two radii

y (units)

x (units)

Highlight a line or construction
  • Circle · Every boundary point is the same distance from its centre.
  • Diameter = 8 · Through the centre.
  • Radius = 4 · Half the diameter.
  1. OO · (0, 0)
  2. 2r = 4 · (0, 4)

An ideal circle; all lengths are in the same unit. Equal units on both axes.

Read the plotted values
Circle
xy
40
3.96577944549524150.5221047688802063
3.86370330515627321.035276180410083
3.6955181300451471.5307337294603591
3.4641016151377551.9999999999999998
3.17341336116494072.4350457160348826
2.82842712474619032.82842712474619
2.43504571603488263.1734133611649407
2.00000000000000043.4641016151377544
1.53073372946035933.695518130045147
1.0352761804100833.8637033051562732
0.52210476888020683.9657794454952415
2.4492935982947064e-164
-0.52210476888020553.965779445495242
-1.03527618041008343.8637033051562732
-1.5307337294603593.695518130045147
-1.99999999999999913.464101615137755
-2.43504571603488263.1734133611649407
-2.828427124746192.8284271247461903
-3.1734133611649412.4350457160348817
-3.4641016151377551.9999999999999998
-3.6955181300451471.5307337294603596
-3.8637033051562731.035276180410084
-3.96577944549524150.5221047688802063
-44.898587196589413e-16
-3.965779445495242-0.5221047688802053
-3.8637033051562737-1.0352761804100814
-3.695518130045148-1.530733729460357
-3.4641016151377544-2.0000000000000004
-3.1734133611649407-2.4350457160348826
-2.8284271247461907-2.82842712474619
-2.4350457160348835-3.1734133611649398
-2.0000000000000018-3.464101615137754
-1.530733729460358-3.6955181300451474
-1.0352761804100825-3.8637033051562732
-0.5221047688802065-3.9657794454952415
-7.347880794884119e-16-4
0.5221047688802085-3.9657794454952415
1.0352761804100845-3.863703305156273
1.53073372946036-3.6955181300451465
2.0000000000000004-3.4641016151377544
2.435045716034882-3.1734133611649407
2.8284271247461894-2.8284271247461907
3.1734133611649398-2.4350457160348835
3.4641016151377535-2.0000000000000018
3.6955181300451474-1.5307337294603582
3.8637033051562732-1.0352761804100827
3.9657794454952415-0.5221047688802067
4-9.797174393178826e-16
Diameter = 8
xy
-40
40
Radius = 4
xy
00
04
Circumference
A circle’s boundary length: C = πd = 2πr.
π
The exact constant C/d, irrational and approximately 3.14159.

Doubling a circle’s radius doubles its circumference. π does not change with circle size. 22/7 and 3.14 are approximations, so use ≈ when rounding.

The chapter describes measurement and polygon methods from several mathematical traditions. More polygon sides improve a bound; a finite approximation is still not the exact π.

Let’s look a little closer.

Go deeper: Measure to estimate, do not declare an exact value

Wrapping a thin thread around a circular object and measuring circumference and diameter estimates their ratio. Errors in thread thickness and ruler readings are expected. Geometry explains why the ratio is a universal constant.

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Question

A wheel has radius 0.4 m. How far does one turn cover without slipping?

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NCERT reference: chapter PDF pages 3, 4, 5, 6, 7, 8.

03

UNDERSTAND · APPLY · CHECK

An arc takes a fraction of the circumference

Use the central angle to find a curved length.

The arc is only the curved part

y (units)

x (units)

Highlight a line or construction
  • Circle · Every boundary point is the same distance from its centre.
  • Chosen arc · The fraction θ/360 of the whole boundary.
  • Two radii · These also belong to the sector’s perimeter.
  1. OO · (0, 0)
  2. AA · (6, 0)
  3. BB · (-3, 5.1962)

Radius 6 units; central angle 120°. The chosen arc is the counterclockwise minor arc. Equal units on both axes.

Read the plotted values
Circle
xy
60
5.9486691682428620.7831571533203094
5.795554957734411.5529142706151244
5.543277195067722.2961005941905386
5.1961524227066322.9999999999999996
4.7601200417474113.652568574052324
4.2426406871192864.242640687119285
3.6525685740523244.760120041747411
3.0000000000000015.196152422706632
2.2961005941905395.54327719506772
1.55291427061512445.79555495773441
0.78315715332031035.948669168242862
3.6739403974420594e-166
-0.78315715332030835.948669168242863
-1.5529142706151255.79555495773441
-2.29610059419053865.54327719506772
-2.99999999999999875.196152422706632
-3.6525685740523244.760120041747411
-4.2426406871192854.242640687119286
-4.7601200417474113.6525685740523226
-5.1961524227066322.9999999999999996
-5.543277195067722.2961005941905395
-5.7955549577344091.5529142706151262
-5.9486691682428620.7831571533203094
-67.347880794884119e-16
-5.948669168242863-0.7831571533203079
-5.79555495773441-1.5529142706151222
-5.543277195067722-2.2961005941905355
-5.196152422706632-3.000000000000001
-4.760120041747411-3.652568574052324
-4.242640687119286-4.242640687119285
-3.6525685740523253-4.76012004174741
-3.0000000000000027-5.196152422706631
-2.296100594190537-5.543277195067721
-1.5529142706151238-5.79555495773441
-0.7831571533203098-5.948669168242862
-1.102182119232618e-15-6
0.7831571533203128-5.948669168242862
1.5529142706151267-5.795554957734409
2.29610059419054-5.54327719506772
3.000000000000001-5.196152422706632
3.6525685740523235-4.760120041747411
4.242640687119284-4.242640687119286
4.76012004174741-3.6525685740523253
5.19615242270663-3.0000000000000027
5.543277195067721-2.2961005941905372
5.79555495773441-1.552914270615124
5.948669168242862-0.7831571533203101
6-1.4695761589768238e-15
Chosen arc
xy
60
5.9942893294911470.261716324192016
5.9771681885504740.522934456485949
5.9486691682428620.7831571533203094
5.9088465180732481.041889066001582
5.85777604271961.2986376836286173
5.795554957734411.5529142706151244
5.7223017044893621.8042347970256387
5.6381557247154512.0521208599540124
5.543277195067722.2961005941905386
5.4378467222198992.5357095704441965
5.32206499906933052.7704916794102035
5.1961524227066322.9999999999999996
5.0603486748773143.223797650080943
4.9149122657339513.4414586181062763
4.7601200417474113.652568574052324
4.5962666587138683.8567256581192355
4.4236640208607454.053541245693961
4.2426406871192864.242640687119285
4.0535412456939614.423664020860745
3.85672565811923644.596266658713868
3.6525685740523244.760120041747411
3.44145861810627634.914912265733951
3.2237976500809435.060348674877313
3.0000000000000015.196152422706632
2.77049167941020445.32206499906933
2.53570957044419655.437846722219899
2.2961005941905395.54327719506772
2.0521208599540135.63815572471545
1.80423479702563985.722301704489361
1.55291427061512445.79555495773441
1.29863768362861755.8577760427196
1.04188906600158255.908846518073248
0.78315715332031035.948669168242862
0.52293445648594885.977168188550474
0.26171632419201745.994289329491147
3.6739403974420594e-166
-0.261716324192015355.994289329491147
-0.52293445648594945.977168188550474
-0.78315715332030835.948669168242863
-1.04188906600158185.908846518073248
-1.29863768362861665.8577760427196
-1.5529142706151255.79555495773441
-1.80423479702563765.722301704489362
-2.05212085995401245.638155724715451
-2.29610059419053865.54327719506772
-2.5357095704441965.4378467222199
-2.77049167941020265.322064999069331
-2.99999999999999875.196152422706632
Two radii
xy
60
00
-2.99999999999999875.196152422706632
Arc length
L = (θ/360) × 2πr when θ is in degrees.

A 120° arc is one-third of a full circle. With r = 6 cm, its length is (120/360) × 12π = 4π cm.

A sector’s perimeter includes its two radii as well: L + 2r. A semicircle’s curved length is πr, but its entire boundary is πr + 2r.

  1. 1

    Identify the central angle and radius.

  2. 2

    Find the fraction θ/360.

  3. 3

    Multiply by the full circumference.

  4. 4

    For sector perimeter, add the two radii.

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Question

What is the whole perimeter of that 120° sector?

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NCERT reference: chapter PDF pages 8, 9, 10, 11, 12, 13.

04

UNDERSTAND · APPLY · CHECK

Longer outer curves need a head start

Count curved and straight boundary pieces separately.

The two curves together make a circle

Vertical position (m)

Horizontal position (m)

Highlight a line or construction
  • Inner running line · Two straights of length 8 plus two semicircles of radius 2.
  • Outer running line · Same straights; semicircle radius 3.

Idealised one-lap model with equal straight lengths. These are illustrative running lines, not official race specifications. Equal units on both axes.

Read the plotted values
Inner running line
Horizontal positionVertical position
1.2246467991473532e-162
-0.390180644032255941.961570560806461
-0.76536686473017951.8477590650225735
-1.11114046603920461.6629392246050902
-1.4142135623730951.4142135623730951
-1.66293922460509071.1111404660392044
-1.84775906502257350.7653668647301798
-1.96157056080646090.39018064403225633
-22.4492935982947064e-16
-1.961570560806461-0.39018064403225583
-1.847759065022574-0.7653668647301785
-1.6629392246050905-1.1111404660392046
-1.4142135623730954-1.414213562373095
-1.111140466039206-1.6629392246050896
-0.765366864730179-1.8477590650225737
-0.39018064403225733-1.9615705608064606
-3.6739403974420594e-16-2
8-2
8.390180644032256-1.9615705608064609
8.76536686473018-1.8477590650225733
9.111140466039204-1.662939224605091
9.414213562373094-1.4142135623730954
9.66293922460509-1.1111404660392044
9.847759065022574-0.7653668647301791
9.96157056080646-0.39018064403225744
10-4.898587196589413e-16
9.961570560806460.3901806440322565
9.8477590650225740.7653668647301782
9.6629392246050921.1111404660392037
9.4142135623730961.4142135623730934
9.1111404660392041.6629392246050905
8.765366864730181.8477590650225737
8.3901806440322581.9615705608064606
82
02
Outer running line
Horizontal positionVertical position
1.8369701987210297e-163
-0.58527096604838392.9423558412096917
-1.14805029709526932.77163859753386
-1.6667106990588072.4944088369076356
-2.12132034355964242.121320343559643
-2.4944088369076361.6667106990588065
-2.771638597533861.1480502970952697
-2.94235584120969130.5852709660483845
-33.6739403974420594e-16
-2.9423558412096917-0.5852709660483837
-2.771638597533861-1.1480502970952677
-2.4944088369076356-1.666710699058807
-2.121320343559643-2.1213203435596424
-1.6667106990588088-2.4944088369076343
-1.1480502970952684-2.7716385975338604
-0.585270966048386-2.942355841209691
-5.51091059616309e-16-3
7.999999999999999-3
8.585270966048386-2.9423558412096913
9.14805029709527-2.77163859753386
9.666710699058806-2.4944088369076365
10.121320343559642-2.121320343559643
10.494408836907635-1.6667106990588065
10.77163859753386-1.1480502970952686
10.942355841209691-0.5852709660483861
11-7.347880794884119e-16
10.9423558412096910.5852709660483848
10.7716385975338621.1480502970952673
10.4944088369076361.6667106990588056
10.1213203435596452.12132034355964
9.6667106990588072.4944088369076356
9.1480502970952692.7716385975338604
8.5852709660483862.942355841209691
8.0000000000000023
03
Stagger
A difference in starting positions that compensates for unequal curve lengths.

In an ideal track made of two equal straight sections and two semicircles, the curves total one full circumference. Length = 2s + 2πr.

Increasing the running radius by Δr adds 2πΔr per lap if the straight lengths stay the same. The smaller track need not have a smaller stagger for the same lane spacing. Actual running-line conventions also matter.

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Question

In our ideal one-lap model, what extra distance comes from increasing the radius by 1 m?

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NCERT reference: chapter PDF pages 1, 9, 10, 11.

05

UNDERSTAND · APPLY · CHECK

Use the perpendicular height

Explain base × height by rearranging the region.

Shear changes the shape, not bh

y (units)

x (units)

Highlight a line or construction
  • Boundary · The sides define the region.
  • Height · Perpendicular to the base.
  1. 1Base = 6 · (3, 0)
  2. 2Height = 3 · (2, 1.5)

Base 6 units and height 3 units give area 18 square units. Equal units on both axes.

Read the plotted values
Boundary
xy
00
60
83
23
00
Height
xy
20
23
Perpendicular height
The shortest distance between the parallel base lines.

Cut a triangle off one side of a parallelogram and move it to the other side. It becomes a rectangle with the same base and perpendicular height, so area = bh.

A sloping side is not usually the height. Two parallelograms can have the same side lengths but different angles, and therefore different areas.

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Question

A parallelogram has base 8 cm, sloping side 5 cm and perpendicular height 3 cm. What is its area?

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NCERT reference: chapter PDF pages 13, 14, 15.

06

UNDERSTAND · APPLY · CHECK

A triangle is half a parallelogram

Explain ½bh, including an outside altitude.

An altitude can fall outside the triangle

y (units)

x (units)

Highlight a line or construction
  • Boundary · The sides define the region.
  • Base extended · Extend the base to meet the altitude.
  • Perpendicular height · Measure straight down, not along the sloping side.
  1. BB · (0, 0)
  2. CC · (6, 0)
  3. AA · (-2, 3)
  4. FootFoot · (-2, 0)

Base BC = 6 units, perpendicular height = 3 units; area = 9 square units. Equal units on both axes.

Read the plotted values
Boundary
xy
00
60
-23
00
Base extended
xy
-20
00
Perpendicular height
xy
-20
-23
Triangle area
A = ½ × base × perpendicular height.
Median
A segment from a vertex to the opposite side’s midpoint.

Two congruent copies of a triangle fit into a parallelogram. Each takes half its area. The formula still works when the altitude meets the extended base.

A median divides the base into equal lengths. The two smaller triangles have the same height, so they have equal areas; they need not be congruent.

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Question

Does equal area imply two triangles have the same shape?

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NCERT reference: chapter PDF pages 15, 16, 17, 25, 26.

07

UNDERSTAND · APPLY · CHECK

When height is missing, use the three sides

Calculate an area from side lengths with Heron’s formula.

Three sides give the area

y (units)

x (units)

Highlight a line or construction
  • Boundary · The sides define the region.
  • Height 12 · A useful check after Heron’s calculation.
  1. AA · (0, 0)
  2. BB · (14, 0)
  3. CC · (5, 12)

AB = 14, AC = 13, BC = 15 units. The model is drawn to scale. Equal units on both axes.

Read the plotted values
Boundary
xy
00
140
512
00
Height 12
xy
50
512
Semiperimeter s
Half the perimeter: (a+b+c)/2.
Heron’s formula
Area = √[s(s−a)(s−b)(s−c)] for a valid triangle.

For sides 13, 14, 15 units, s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 square units.

Check the triangle inequality first: every pair of sides must sum to more than the third. Keep the area unit squared, and do not mistake s for the perimeter.

  1. 1

    Check that the lengths form a triangle.

  2. 2

    Compute s = (13 + 14 + 15)/2 = 21.

  3. 3

    Find s−a, s−b, s−c: 8, 7, 6.

  4. 4

    Multiply and take the square root: 84 units².

PAUSE & TRY · think first, then check with a free account

Question

Can sides 2, 3, 6 form a triangle to which we apply Heron’s formula?

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NCERT reference: chapter PDF pages 17, 18, 19, 25, 32.

CHAPTER RECAP

Perimeter follows the boundary → Every circle has the same C/d ratio → An arc takes a fraction of the circumference → Longer outer curves need a head start → Use the perpendicular height → A triangle is half a parallelogram → When height is missing, use the three sides

Learnijoy explanations and visuals based on NCERT Class IX Mathematics · Chapter 6 ↗. Exploring visuals and revealing answers do not record completion.

The rest of this chapter

Keep reading Measuring Space: Perimeter and Area, free

  1. Locked: 1. Circle radii give two other area connections
  2. Locked: 2. Four sides need the cyclic condition
  3. Locked: 3. Same area does not mean the same perimeter
  4. Locked: 4. Rearrange slices to see πr²
  5. Locked: 5. The same angle fraction also divides area
  6. Locked: 6. Subtract the triangle from the sector
  7. Locked: 7. Length scales once; area scales twice

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