CLASS 9 • MATHS • CHAPTER 6
Measuring Space: Perimeter and Area
Short, colourful, handwritten-style notes by EduGrown
1What is Perimeter?
📝 Perimeter
The perimeter of a shape is the total length around its border. Imagine a tiny insect walking around the border without turning back — the distance it travels is the perimeter.
⭐ Basic perimeters
- Square of side a → perimeter = 4a
- Equilateral triangle of side a → perimeter = 3a
- Rectangle of length a, width b → perimeter = 2(a + b)
💡 The big idea behind π
For every square, perimeter : side = 4 : 1. For every equilateral triangle it is 3 : 1. The ratio never changes with size — it depends only on the shape. So a circle must also have its own fixed ratio!
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2Perimeter of a Circle — the C/D Ratio
📝 Circumference
The perimeter of a circle has a special name — the circumference (C). The ratio of circumference to diameter, C/D, is the same for circles of every size. We call this constant π (pi).
⭐ The two most-used formulas
C = πD and C = 2πr
since diameter D = 2 × radius r.
✏️ Try it at home
Take a cotton reel. Measure its diameter D. Wrap thin thread tightly around it 20 times, unwrap and measure the length L. Now compute L20 D. You will get a number between 3.1 and 3.2 — that is π!
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3The Story of π — Quick Timeline
| Who & When | Value of π |
|---|---|
| Mesopotamia (c. 1900 BCE) | 3 + 18 = 3.125 |
| Archimedes (250 BCE) | 31071 < π < 317 |
| Ptolemy (150 CE) | 377120 ≈ 3.14167 |
| Zu Chongzhi (480 CE) | 355113 ≈ 3.1415929 |
| Āryabhaṭa (499 CE) | 3.1416 (called it ásanna = approximate) |
| Brahmagupta (628 CE) | √10 ≈ 3.1622 |
| Mādhava (c. 1400) | First exact formula (infinite series) |
📜 Mādhava’s beautiful formula
π4 = 1 − 13 + 15 − 17 + …
This infinite series gave π correct to 11 decimal places and gave birth to the branch of maths called calculus.
💡 Why the symbol π?
In 1706, William Jones used the Greek letter π because it is the first letter of the Greek word perimetros (perimeter). Euler made it popular — and we still use it today.
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4π is Irrational
📝 Irrational number
A number that cannot be written as ab (a, b integers, b ≠ 0) is called irrational. Fractions give repeating decimals, but π has no pattern at all.
Lambert proved π is irrational in 1761.
Lambert proved π is irrational in 1761.
⚠️ Close, but NOT equal
Always write π ≈ 227, never π = 227. Same for √2 ≈ 1.414. Since π is irrational, there is no ‘best fraction’ for it — a closer one always exists.
✏️ Fun way to remember π
“How I wish I could recollect pi”
Count the letters in each word: 3, 1, 4, 1, 5, 9, 2 → π ≈ 3.141592
🎉 14 March (3-14) is Pi Day and 22 July (22-7) is Pi Approximation Day.
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Count the letters in each word: 3, 1, 4, 1, 5, 9, 2 → π ≈ 3.141592
🎉 14 March (3-14) is Pi Day and 22 July (22-7) is Pi Approximation Day.
5Length of an Arc of a Circle
⭐ Arc lengths
- Semicircle = 2πr ÷ 2 = πr (also 2πr × 180°360°)
- Quarter circle = 2πr ÷ 4 = πr2 (also 2πr × 90°360°)
⭐ General arc formula
If arc AB subtends an angle θ° at the centre of a circle of radius r:
Arc length = 2πr × θ°360°
Simple idea: an arc is just a fraction of the full circle, and that fraction is θ360.
6The 400 m Athletics Track & Stagger
- Two straight sections of 84.39 m each → 168.78 m
- Two semicircles of radius 36.8 m → together they make one full circle
- Circle’s circumference = 2 × 3.1416 × 36.8 = 231.22 m
- Total = 168.78 + 231.22 = 400 m ✔
💡 Why do runners start at different points?
On the straight parts everyone runs equal distance. But on the curves, an outer lane has a bigger radius, so a longer arc. The head-start given to outer lanes to balance this is called the stagger.
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7Area of Rectangle & Parallelogram
⭐ Remember
- Rectangle of sides a, b → Area = ab sq. units
- Square of side a → Area = a2 sq. units
⭐ Area of a parallelogram
Area = base × height = b h
The rectangle formed has the same base and same height, so the two shapes (though different) have equal area.
⚠️ Careful!
For a rectangle, knowing the sides is enough to find the area. For a parallelogram it is NOT — keep the sides fixed and squash it, the area keeps changing. You must know the height.
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8Area of a Triangle
⭐ The formula
Area of triangle = 12 × base × height = 12 b h
💡 The neat proof
Two congruent copies of a triangle fit together to form a parallelogram of the same base and height.
So triangle = 12 of parallelogram = 12 bh ✔
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So triangle = 12 of parallelogram = 12 bh ✔
9Median Divides a Triangle Equally
📝 Median
A median is the line segment joining a vertex to the midpoint of the opposite side.
⭐ Theorem
A median of a triangle divides it into two triangles of equal area.
Why? ΔABD and ΔACD have equal bases (BD = DC) and the same height h. So both have area ah2.
Why? ΔABD and ΔACD have equal bases (BD = DC) and the same height h. So both have area ah2.
💡 Surprising!
The two triangles are usually not congruent — they look completely different — yet their areas are exactly equal.
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10Heron’s Formula
⭐ Area from the three sides only
If ΔABC has sides a, b, c, first find the semi-perimeter:
s = 12(a + b + c)
Area = √( s(s − a)(s − b)(s − c) )
Use it when the height is not given.
✏️ Quick check: sides 3, 4, 5
s = 12(3 + 4 + 5) = 6
Area = √(6 × 3 × 2 × 1) = √36 = 6 sq. units
Check: 3² + 4² = 5², so it is right-angled → 12 × 3 × 4 = 6 ✔ Same answer!
Area = √(6 × 3 × 2 × 1) = √36 = 6 sq. units
Check: 3² + 4² = 5², so it is right-angled → 12 × 3 × 4 = 6 ✔ Same answer!
| Triangle | Area by Heron’s formula |
|---|---|
| Equilateral, side a | √34 a2 |
| Isosceles, equal sides a, base 2b | b √(a2 − b2) |
| Sides 3, 4, 5 | 6 sq. units |
📜 Who was Heron?
A Greek mathematician and inventor who taught at the Museum in Alexandria, ancient Egypt, on the banks of the Nile.
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11Two More Triangle-Area Formulas
📝 Two special circles
- Circumcircle — passes through all three vertices. Radius = R.
- Incircle — fits tightly inside, touching all three sides. Radius = r.
⭐ Beautifully symmetric formulas
Area of ΔABC = abc4R
Area of ΔABC = r(a + b + c)2 = r × s
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12Brahmagupta’s Formula (Cyclic 4-gon)
⚠️ Sides alone are not enough
Knowing only the four sides of a 4-gon does not fix its area. We need one extra piece of information — an angle, a diagonal, or a special property such as being cyclic.
⭐ Brahmagupta’s formula (628 CE)
For a cyclic 4-gon with sides a, b, c, d and s = 12(a + b + c + d):
Area = √( (s − a)(s − b)(s − c)(s − d) )
💡 Heron is hidden inside Brahmagupta!
Put d = 0 (the fourth side shrinks to nothing, so the 4-gon becomes a triangle). Then s becomes 12(a + b + c) and the formula turns into
√( s(s − a)(s − b)(s − c) )
which is exactly Heron’s formula. So Brahmagupta’s formula is a generalisation of Heron’s.
✏️ Check with a rectangle
Rectangle with sides a, b, a, b → s = a + b.
Area = √(b · a · b · a) = ab ✔ Correct!
Area = √(b · a · b · a) = ab ✔ Correct!
13Squaring a Rectangle
📝 What does it mean?
To ‘square a shape’ means to construct a square of exactly the same area as that shape — using only geometry.
📜 Baudhāyana’s construction (800 BCE)
In the Śhulbasūtra, Baudhāyana showed how to square a rectangle with sides a and b. The whole construction is really a picture of the algebra identity:
(a + b2)2 − (a − b2)2 = ab
And ab is exactly the area of the rectangle. Beautiful!
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14Area of a Circle
💡 Archimedes’ thought experiment
Keep increasing the number of sides of a regular polygon — it gets closer and closer to a circle. Applying the polygon rule:
Area = 12 × circumference × radius = 12 × 2πr × r = πr2
⭐ The easiest visual proof
Cut the circle into thin slices and rearrange them. As slices get thinner, the shape becomes a parallelogram with
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- base = half the circumference = πr
- height = radius = r
15Area of a Sector & Segment
📝 Sector vs Segment
- Sector = region between an arc and the two radii at its ends (pizza-slice shape).
- Segment = region between an arc and the chord joining its ends.
⭐ Sector formula
If the sector angle is θ°:
Area of sector = πr2 × θ°360°
Perimeter of sector = 2πr × θ°360° + 2r
(arc + the two straight radii)
💡 One rule for everything
Both the arc formula and the sector formula come from the same simple idea: take the whole circle and keep the fraction θ360 of it.
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16All Formulas at a Glance
| Shape | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a2 |
| Rectangle (a, b) | 2(a + b) | ab |
| Parallelogram | 2(a + b) | base × height |
| Triangle | a + b + c | 12 bh |
| Trapezium | sum of sides | 12(a + b)h |
| Circle (radius r) | 2πr | πr2 |
| Semicircle | πr + 2r | 12πr2 |
| Quarter circle | πr2 + 2r | 14πr2 |
| Sector (angle θ°) | 2πr·θ360 + 2r | πr2·θ360 |
⭐ The two ‘side-only’ formulas
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- Heron (triangle): √( s(s−a)(s−b)(s−c) ), s = a+b+c2
- Brahmagupta (cyclic 4-gon): √( (s−a)(s−b)(s−c)(s−d) ), s = a+b+c+d2
- Triangle via circles: abc4R and r(a+b+c)2
17Common Mistakes to Avoid
❌ Don’t do this
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- Writing π = 227. It is only ≈. π is irrational.
- Mixing up radius and diameter — C = 2πr but C = πD. Read the question carefully!
- Forgetting the 2 straight radii while finding the perimeter of a sector (arc alone is not the perimeter).
- Using the slant side as the height in a parallelogram or triangle. Height must be perpendicular to the base.
- Thinking the sides alone decide the area of a parallelogram or a 4-gon. They do not.
- Applying Brahmagupta’s formula to any 4-gon. It works only when the 4-gon is cyclic.
- In Heron’s formula, using the full perimeter instead of the semi-perimeter s.
- Forgetting units: perimeter in cm/m, area in cm2/m2.
18Chapter Summary — Quick Revision
🏆 All of Chapter 6 in 12 lines
- Perimeter = total length around a shape; area = space it covers, measured in unit squares.
- For any shape, perimeter : side stays fixed — for a circle this fixed ratio C : D is π.
- C = 2πr = πD, and π ≈ 227 ≈ 3.14.
- π is irrational — its decimals never end and never repeat. Mādhava gave the first exact formula for it.
- Arc length = 2πr × θ360; sector area = πr2 × θ360.
- A 400 m track = 2 straights + 2 semicircles; outer lanes need a stagger because their curves are longer.
- Rectangle = ab; parallelogram = base × height; triangle = 12 bh.
- A median splits a triangle into two triangles of equal area (even though they look different).
- Heron’s formula gives the area of a triangle from its three sides alone.
- Brahmagupta’s formula does the same for a cyclic 4-gon — and Heron’s formula is its special case (d = 0).
- Area of a circle = πr2, proved by slicing the circle and rearranging it into a parallelogram.
- Half disc = 12πr2, quarter disc = 14πr2 — always just a fraction of the full circle.
✨ Happy Learning — @edugrown ✨