📐 Chapter 7: Area
Complete Study Notes — Formulae, Concepts, Examples & Practice Questions
What is Area? — The Big Idea
The area of a region is measured by counting how many unit squares (squares of side 1 unit) can fit inside it. Each unit square has area 1 unit².
Think of filling a shape with rangoli powder. The more powder needed, the larger the area! The rectangle with sides 7 cm and 4 cm needs more powder than one with sides 8 cm and 3 cm — even though both look similar in size.
📌 Why Not Use Perimeter as a Measure of Area?
The perimeter (boundary length) of a shape does NOT tell us its area. Two shapes can have the same perimeter but different areas, and vice versa!
A bigger perimeter does NOT mean a bigger area. Example: A very flat, long rectangle can have a huge perimeter but tiny area.
Rectangle & Square
The area of a rectangle equals the number of unit squares that can be packed inside it — which is just the product of its two side lengths.
✏️ Worked Example
Area = 7 × 4 = 28 cm²Rectangle 2: Length = 8 cm, Width = 3 cm
Area = 8 × 3 = 24 cm²∴ Rectangle 1 has the larger area (needs more rangoli powder!)
🔺 Triangle from a Rectangle
The diagonal of a rectangle divides it into two congruent triangles. Each triangle has half the area of the rectangle.
For the 7 cm × 4 cm rectangle, each triangle formed by its diagonal has area = ½ × 7 × 4 = 14 cm².
🧩 Dividing a Square into Equal Parts
A square can be divided into 4 equal-area parts in infinitely many ways! You can start with a simple 4-quadrant division and then modify each part — as long as area gained equals area lost, all 4 parts remain equal.
Cutting a figure into pieces and rearranging them to form a new figure of the same area is called dissection.
Area of a Triangle
To find the area of any triangle, we need its base and its height (also called altitude — the perpendicular distance from the base to the opposite vertex).
📌 Understanding Base and Height
- The base is any side of the triangle.
- The height is the perpendicular drawn from the opposite vertex to the base (or its extension).
- Any side can be chosen as the base — the area will always be the same.
✏️ How the Formula is Derived
- Draw a triangle ABC with base BC.
- Draw the altitude AX perpendicular to BC.
- Extend a line through A parallel to BC, and extend the perpendiculars — this forms rectangle BCDE.
- The triangle ABC exactly fills half of this rectangle.
- So, Area(△ABC) = ½ × BC × AX = ½ × base × height.
This formula works for ALL types of triangles — acute, right-angled, and obtuse. For an obtuse triangle, the height falls outside the triangle (on the extended base), but the formula still holds.
✏️ Worked Examples
Area = ½ × 4 × 3 = 6 cm²Example 2: Base (EF) = 5 cm, Altitude (EN) = 3.2 cm
Area = ½ × 5 × 3.2 = 8 cm²Example 3: Base (AT) = 3 cm, Height (NA) = 4 cm
Area = ½ × 3 × 4 = 6 cm²
🔍 Useful Property: Finding an Unknown Altitude
If you know the area of a triangle and one base-height pair, you can find another altitude using the same area.
Given: △ABC, AX = 3 units, BC = 5 units, AC = 4 unitsArea(△ABC) = ½ × AX × BC = ½ × 3 × 5 = 15/2 sq. units
Also: Area = ½ × BY × AC = ½ × BY × 4 = 2·BY
So: 2·BY = 15/2 → BY = 3.75 units
🔁 Triangles Between Parallel Lines (Same Base)
If multiple triangles share the same base BC and their third vertices all lie on a line parallel to BC, then all these triangles have equal area — because they all have the same base and the same height (distance between the parallel lines).
Among all triangles with a common base BC and the third vertex on a fixed parallel line, the triangle with the minimum perimeter is the isosceles one (vertex on the perpendicular bisector of BC). This is proven using mirror reflections!
📌 Key Property — Median Divides Triangle Equally
In a triangle, the line joining a vertex to the midpoint of the opposite side (called a median) divides the triangle into two triangles of equal area.
Ancient Indian geometric texts called the Śulba-Sūtras (used for building altars) contain many problems about transforming one shape into another of equal area — like converting a rectangle into a triangle, or a triangle into a rectangle! The same problems appear in Euclid’s Elements from ancient Greece.
Area of any Polygon
Any polygon — whether it is a quadrilateral, pentagon, hexagon, or any other shape — can be divided into triangles by drawing diagonals from one vertex.
📐 Quadrilateral ABCD
Join diagonal BD. This splits ABCD into △ABD and △BCD. Measure the base (BD) and heights from A and C respectively.
AC = 22 cm, BM ⊥ AC (BM = 3 cm), DN ⊥ AC (DN = 3 cm)Area(△ABC) = ½ × AC × BM = ½ × 22 × 3 = 33 cm²
Area(△ACD) = ½ × AC × DN = ½ × 22 × 3 = 33 cm²
Total Area(ABCD) = 33 + 33 = 66 cm²
For a quadrilateral with diagonal d and perpendiculars h₁ and h₂ from the other two vertices:
Area = ½ × d × (h₁ + h₂)
📌 Area of a Regular Hexagon
To find the area of a regular hexagon, you need only one measurement — the side length. The hexagon can be split into 6 equilateral triangles, all of equal area.
Parallelogram
A parallelogram can be converted into a rectangle of equal area by dissection. Cut a triangle from one end and attach it to the other.
- Start with parallelogram ABCD.
- Draw AX perpendicular to DC (this is the height).
- Cut off △AXD.
- Move it to the right to fill the gap — ABCX becomes rectangle ABYX.
- Both shapes have equal area!
The height must be perpendicular to the chosen base. Any side can be the base — the corresponding perpendicular height is used. Same area no matter which base you choose!
✏️ Worked Examples
Base = 7 cm, Height = 4 cm
Area = 7 × 4 = 28 cm²
Base = 5 cm, Height = 3 cm
Area = 5 × 3 = 15 cm²
Base = 4.8 cm, Height = 5 cm
Area = 4.8 × 5 = 24 cm²
Base = 4.4 cm, Height = 2 cm
Area = 4.4 × 2 = 8.8 cm²
📌 Rectangle vs Parallelogram (Same Side Lengths)
A rectangle with sides 5 cm and 4 cm has area = 20 cm². A parallelogram with the same side lengths but slanted always has a smaller height than 4 cm, so its area is always less than 20 cm². The rectangle always has the greater area!
In a parallelogram, the slanted side is NOT the height. Always use the perpendicular distance between the parallel sides.
✏️ Finding a Missing Height — QN
PQ = 7.6 cm (one side), PS = 12 cm (base)
QM (height on base PS) = 6 cmArea = PS × QM = 12 × 6 = 72 cm²
Also: Area = PQ × QN → 7.6 × QN = 72
∴ QN = 72 ÷ 7.6 ≈ 9.47 cm
Rhombus
A rhombus is a parallelogram with all sides equal. So we can use Area = base × height. But it also has a special property — its diagonals are perpendicular bisectors of each other — giving us a much simpler formula!
🔑 Deriving the Diagonal Formula
- Diagonals AC and BD bisect each other at right angles at point O.
- The rhombus is split into 4 right-angle triangles.
- Rearrange the 4 triangles to form a rectangle of width = AC and height = BD/2.
- Area of rectangle = AC × BD/2.
- Therefore, Area of rhombus = ½ × AC × BD.
where d₁ and d₂ are the lengths of the two diagonals
You can also find the area of a rhombus using the triangle method:
Area = Area(△ADB) + Area(△CDB) = ½ × AO × BD + ½ × CO × BD = ½ × AC × BD ✓
✏️ Worked Example
= ½ × 300
= 150 cm²
Split the rhombus into two isosceles triangles (△ABD and △CBD). Convert each into a rectangle. Stack the two rectangles — the result is a single rectangle with the same area as the rhombus. This elegant method comes from ancient India!
Trapezium
A trapezium is a quadrilateral with exactly one pair of parallel sides. These parallel sides are called the parallel sides (or bases), and the distance between them is the height.
🔑 Derivation — Breaking into Rectangle and Triangles
- Let WXYZ be a trapezium with WX ∥ ZY (parallel sides = a and b, height = h).
- Draw WM ⊥ ZY and XN ⊥ ZY. This creates rectangle WXNM in the middle.
- Two right triangles (△WMZ and △XNY) appear on the sides.
- Area = ½hx + ha + ½hy = h(½x + a + ½y)
- Since b = x + y + a, we get x + y = b − a.
- Area = ½h(b − a + 2a) = ½h(a + b). ✓
🔑 Alternative Method — Two Copies of Trapezium
Take two identical copies of the trapezium. Rotate one 180° and join them along the non-parallel side. The result is a parallelogram with base (a + b) and height h.
✏️ Worked Examples
Parallel sides: 24 m and 36 m
Height: 14 m
Area = ½ × 14 × (24 + 36)
= ½ × 14 × 60 = 420 m²
Parallel sides: 6 in and 14 in
Height: 10 in
Area = ½ × 10 × (6 + 14)
= ½ × 10 × 20 = 100 in²
Parallel sides: 12 ft and 18 ft
Height: 8 ft
Area = ½ × 8 × (12 + 18)
= ½ × 8 × 30 = 120 ft²
(Parallelogram shape)
Base = 16 ft, Height = 7 ft
Area = 16 × 7 = 112 ft²
A regular hexagon can be divided into 1 trapezium + 1 equilateral triangle + 1 rhombus. Their areas are in the ratio 2 : 1 : 1. (The trapezium has twice the area of the other two.)
Areas in Real Life
Area is used everywhere — from flooring your room to measuring agricultural land. Here are the standard conversions you need to know.
| Unit | Equivalent | Use Case |
|---|---|---|
| 1 in (inch) | 2.54 cm | Furniture dimensions |
| 1 ft (foot) | 12 in = 30.48 cm | Room size, furniture |
| 1 in² | 2.54² = 6.4516 cm² | Converting area units |
| 1 ft² | 144 in² | Floor area |
| 1 acre | 43,560 ft² | Agricultural land |
| 1 km² | 1,000,000 m² | City/district area |
✏️ Unit Conversion Examples
5 in = 5 × 2.54 = 12.70 cm
7.4 in = 7.4 × 2.54 = 18.796 cmLengths in inches:
5.08 cm = 5.08 ÷ 2.54 = 2 in
11.43 cm = 11.43 ÷ 2.54 = 4.5 inArea conversion:
10 in² = 10 × 6.4516 = 64.516 cm²
161.29 cm² = 161.29 ÷ 6.4516 = 25 in²
A4 Sheet:
21 cm × 29.7 cm = 623.7 cm²
Different regions of India use their own traditional units: bigha, gaj, katha, dhur, cent, ankanam. Ask your family what unit is used in your region!
The largest city by area in India is Jaisalmer (Rajasthan) — covering about 38,401 km²! In the world, it’s Hulunbuir in China. The smallest cities by area are tiny island or city-state capitals.
Summary of All Formulas
Area = Length × Width
Area = Side²
Area = ½ × Base × Height
Area = Base × Height
Area = ½ × d₁ × d₂
Area = ½ × h × (a + b)
Divide into triangles. Sum their areas.
Area = ½ × diagonal × (sum of heights)
Key Concepts to Remember
- Area ≠ Perimeter — they are completely different measurements
- Height is always perpendicular to the base
- Triangles between parallel lines with the same base → Equal areas
- Median divides a triangle into two equal-area triangles
- Diagonals of a rectangle create 4 equal-area triangles
Exam Practice Questions
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