📐 Measuring Space: Perimeter & Area
Complete notes with formulas, diagrams, history, and practice questions — made simple for Class 9 students
- Introduction — The Race Track Problem
- 6.1 Perimeter of Basic Shapes
- 6.2 Perimeter of a Circle — The C/D Ratio & History of π
- 6.3 π Is Irrational
- 6.4 Length of an Arc of a Circle
- 6.6 Area of a Rectangle & Square
- 6.7 Area of a Parallelogram
- 6.8 Area of a Triangle & Heron’s Formula
- 6.10 Area of a Circle & Sector
- Chapter Summary
- Important Exam Questions
Introduction — The Race Track Problem
Have you ever watched a 4 × 100 m relay race and noticed that athletes in outer lanes start ahead of those in inner lanes? All of them finish at the same line. Why is this so?
The gap between the starting points of adjacent lanes is called a stagger. Athletes in outer lanes travel a greater distance on the curved parts of the track, so they are given a head start. To calculate the stagger, we need to know the circumference (length around) a circle!
A standard 400 m athletics track has two straight sections of 84.39 m each and two curved semicircular ends. The innermost semicircle has a radius of 36.5 m, and each lane is 1.22 m wide.
6.1 Perimeter of Basic Shapes
Perimeter is the total length around the boundary of a shape. Imagine a tiny insect walking all the way around a shape and returning to its starting point — the total distance it covers is the perimeter.
| Shape | Formula for Perimeter |
|---|---|
| Square (side a) | 4a |
| Rectangle (length a, width b) | 2(a + b) |
| Equilateral Triangle (side a) | 3a |
| Circle (radius r) | 2πr (Circumference) |
The ratio of perimeter to side is fixed for each type of shape — 4:1 for squares, 3:1 for equilateral triangles. This pattern holds for circles too — the ratio of circumference to diameter is always π.
6.2 Perimeter of a Circle — The C/D Ratio & History of π
For any circle, the ratio of its circumference (C) to its diameter (D) is always the same constant, no matter how big or small the circle is. This special constant is called π (pi).
🗺️ The Amazing History of π
Mathematicians across the world have been fascinated by π for thousands of years. Here’s how our understanding evolved:
| Mathematician / Civilisation | Time | Approximation for π |
|---|---|---|
| Mesopotamia (Babylon) | ~1900 BCE | 3 + 1/8 = 3.125 |
| Ancient Egypt (Ahmes Papyrus) | ~1500 BCE | (8/9)² × 4 ≈ 3.1605 |
| Archimedes of Syracuse | ~250 BCE | 3.1408 < π < 3.1429 (using 96-sided polygons) |
| Ptolemy of Alexandria | ~150 CE | 377/120 ≈ 3.14167 |
| Liu Hui (China) | 263 CE | 3.14159 (using 3072 sides) |
| Zu Chongzhi (China) | 480 CE | 355/113 ≈ 3.1415929 🏆 |
| Āryabhaṭa (India) | 499 CE | 62832/20000 = 3.1416 |
| Brahmagupta (India) | 628 CE | √10 ≈ 3.1623 |
| Mādhava of Sangamagrāma (India) | ~1400 CE | Exact infinite series! (11 decimal places) |
He “trapped” the circle between inscribed and circumscribed polygons. By calculating perimeters of polygons with more and more sides (up to 96!), he squeezed in on the value of π from both sides: 3 10/71 < π < 3 1/7.
Take a round object (like a cotton reel). Measure its diameter D. Wrap a thin thread around it 20 times and measure the total length L. Then L / (20D) ≈ π. You should get a value between 3.1 and 3.2!
In 1706, Welsh mathematician William Jones first used the Greek letter π (from the Greek word perimetros). It was later popularised by the great Swiss mathematician Leonhard Euler (pronounced “oiler”). We still use it today!
March 14 (3-14) is celebrated as Pi Day. July 22 (22/7) is celebrated as Pi Approximation Day in India!
6.3 π Is Irrational
The digits of π go on forever with no repeating pattern:
π = 3.14159265358979323846264338327950288…
Compare: 1/3 = 0.33333… (pattern: 3 repeats)
1/7 = 0.142857142857… (pattern: 142857 repeats)
π = 3.14159265… (NO pattern — goes on forever!)
A number that CANNOT be written as a fraction p/q (where p, q are integers, q ≠ 0). π and √2 are irrational numbers. The mathematician Lambert proved π is irrational in 1761.
π is NOT equal to 22/7. They are just approximately equal: π ≈ 22/7. In exams, use 22/7 for π only when the question says so. A much better approximation is 355/113.
Count the letters in each word of: “How I wish I could recollect pi”
How(3) I(1) wish(4) I(1) could(5) recollect(9) pi(2) → 3.141592 — Amazing!
6.4 Length of an Arc of a Circle
An arc is a part of the circumference of a circle. The length of an arc depends on how big an angle it subtends at the centre.
Arc Length = 2πr × 180/360 = πr
Arc Length = 2πr × 90/360 = πr/2
🏃 Back to the Athletics Track!
A 400 m track has two straight sections (84.39 m each = 168.78 m total) and two semicircular ends. Lane 1 athlete runs at 0.3 m from inner border (radius = 36.5 + 0.3 = 36.8 m).
Two semicircles (= 1 full circle, r = 36.8 m):
Circumference = 2 × π × 36.8 = 2 × 3.1416 × 36.8 = 231.22 m
Total distance = 168.78 + 231.22 = 400.00 m ✔
Lane 2 has a larger radius → larger circumference → needs a STAGGER!
Stagger between lane 1 and lane 2 = 2π × (width of lane) = 2π × 1.22 ≈ 7.66 m
6.6 Area of a Rectangle & Square
Area is the amount of 2D space enclosed within a boundary. The unit of area is 1 sq. unit (or 1 unit²).
Length = a, Width = b
Area = a × b = ab sq. units
Side = a
Area = a² sq. units
Among all rectangles with the same perimeter (say 40 units), which one has the largest area? The answer is the square! A square with side 10 gives area 100, while a 5×15 rectangle gives only 75. This is a beautiful result in optimisation!
6.7 Area of a Parallelogram
A parallelogram can be cut and rearranged into a rectangle with the same base and height. So:
You cannot find the area of a parallelogram just from the lengths of its sides — you also need the height. Two parallelograms with the same side lengths can have very different areas (think of squishing one flat).
6.8 Area of a Triangle & Heron’s Formula
Two congruent triangles can be joined to form a parallelogram. So the area of a triangle is half the area of a parallelogram with the same base and height.
🏛️ A Median Divides a Triangle into Equal Areas!
A median connects a vertex to the midpoint of the opposite side. Since both halves share the same height and have equal bases (the midpoint divides the base equally), they have equal areas.
A median of a triangle divides it into two triangles with equal area. Remarkably, these two triangles may be differently shaped (not congruent), yet their areas are the same!
🔢 Heron’s Formula — Area from Side Lengths Alone!
What if you don’t know the height, but you know all three sides? Heron of Alexandria gave us a brilliant formula:
s = (a + b + c) / 2 (semi-perimeter)
Area = √[s(s−a)(s−b)(s−c)]
✅ Worked Examples
✔ Check: 3²+4²=5² → right-angled triangle → Area = ½×3×4 = 6 sq. units ✓
💫 Brahmagupta’s Formula for Cyclic Quadrilaterals
Brahmagupta (628 CE) extended Heron’s formula to cyclic quadrilaterals (4-sided figures inscribed in a circle):
s = (a + b + c + d) / 2
Area = √[(s−a)(s−b)(s−c)(s−d)]
Brahmagupta’s formula is a generalisation of Heron’s formula! If we put d = 0 (shrink one side to zero, turning the 4-gon into a triangle), Brahmagupta’s formula becomes exactly Heron’s formula. This is a powerful example of special cases and generalisation in mathematics.
Unlike a triangle, a quadrilateral with given side lengths can have many different areas — you need extra info (an angle, a diagonal, or the constraint that it’s cyclic).
6.10 Area of a Circle & Sector
📖 Historical Background
Ancient Babylonians (before 1500 BCE) found that C² : A ≈ 12:1, giving them Area ≈ C²/12. Around 1500 BCE, ancient Egyptians found Area ≈ (8d/9)², very close to the true value.
Finally, around 250 BCE, Archimedes proved the exact formula: Area of circle = ½ × circumference × radius.
Slice the circle like a pizza. Rearrange the slices alternately up-and-down. As slices get thinner and thinner, the shape approaches a parallelogram with base = πr (half the circumference) and height = r. Area = base × height = πr × r = πr²!
🥧 Area of a Sector of a Circle
A sector is a “pizza slice” — the region bounded by two radii and an arc.
Area = πr² × 180/360 = πr²/2
Area = πr² × 90/360 = πr²/4
🔲 Area of a Segment
A segment is the region between a chord and the arc it cuts off.
Chapter Summary — All Key Formulas at a Glance
C = 2πr or πd
where r = radius, d = diameter
l = 2πr × θ°/360°
θ = central angle
A = ½ × base × height
= ½ bh sq. units
s = (a+b+c)/2
A = √[s(s−a)(s−b)(s−c)]
A = πr² sq. units
π ≈ 22/7 ≈ 3.14
A = πr² × θ°/360°
θ = central angle
Area = base × height = bh
Area = ½(a+b)×h
a, b = parallel sides, h = height
Cyclic quadrilateral sides a,b,c,d
s=(a+b+c+d)/2
A=√[(s−a)(s−b)(s−c)(s−d)]
π is irrational (non-terminating, non-repeating). π ≈ 22/7 (use in exams unless told otherwise). π ≈ 355/113 (a much better approximation). π ≈ 3.14159265…
Important Exam Questions with Solutions
Area = ½(40+20)×24 = ½×60×24 = 720 cm²
Leave a Reply