AREA
Rectangles & Squares · Triangles · Parallelogram · Rhombus · Trapezium
7.1 Rectangles & Squares
📏 Area of a Rectangle
The area of a region is measured by counting the number of unit squares that fit inside it. A unit square has sidelength 1 unit.
- Rectangle (7 cm × 4 cm) → 7 × 4 = 28 unit squares → Area = 28 cm²
- Rectangle (8 cm × 3 cm) → 8 × 3 = 24 unit squares → Area = 24 cm²
✂️ Key Insight — Dissection
You can divide a square into 4 equal-area parts in infinitely many ways! If you compress one edge and expand another by the same amount in each part, the area stays the same. This idea of cutting and rearranging shapes to get equal areas is called dissection.
Area & Rangoli
🎨 Rangoli & Area
Rangoli uses coloured powder to fill regions of different shapes. The amount of powder needed depends on the area of the region — not its perimeter!
Why Perimeter ≠ Area
🚫 Perimeter is NOT a measure of Area
- Two regions can have the same perimeter but different areas
- Two regions can have the same area but different perimeters
- We can even find Region 1 and Region 2 where: Perimeter₁ > Perimeter₂ but Area₁ < Area₂ !
Triangles
🔑 Key Idea — Triangle = Half a Rectangle
The diagonal of a rectangle divides it into two congruent triangles. So the area of each triangle = half the area of the rectangle.
This is true for any triangle with the same base and height — no matter where the apex (top vertex) is on the parallel line above the base, the area is the same!
Area of a Triangle
📏 How to find the Area
- Draw the triangle with base BC
- Draw a line l through A parallel to BC
- Drop perpendiculars from B and C to line l
- This forms rectangle BCDE with height = altitude of triangle
- Triangle area = ½ × base × height
For an obtuse triangle: Area(△ABC) = Area(△ADC) − Area(△ADB) = ½×h×DC − ½×h×DB = ½×h×BC ✓
Applications of the Triangle Formula
🔍 Finding an Altitude
If you know the area of a triangle, you can find any altitude by using a different base-height pair:
Area = ½ × base₁ × h₁ = ½ × base₂ × h₂
📌 Diagonals of Rectangle
The two diagonals of a rectangle divide it into 4 triangles of equal area — since adjacent triangles share the same base (OB = OD, diagonals bisect each other) and same height.
📐 Triangles Between Parallel Lines
- Line l ∥ BC. All triangles with base BC and apex on line l have the same area (same base, same height)
- The triangle with minimum perimeter has its apex on the perpendicular bisector of BC
- Proof uses the mirror/reflection trick: imagining line l as a mirror, AB = AB′, so B→A→C = B→A→C′. Shortest path = straight line BC′
Area of any Polygon
🔺 The Universal Method
- Any polygon can be divided into triangles by drawing diagonals from one vertex
- Find the area of each triangle using ½ × base × height
- Add all triangle areas → total polygon area!
- A quadrilateral needs 1 diagonal → 2 triangles
- A pentagon needs 2 diagonals → 3 triangles
- An n-gon needs (n−2) triangles
If AC = 22 cm, BM ⊥ AC (BM = 3 cm), DN ⊥ AC (DN = 3 cm):
Parallelogram
✂️ Dissection to Rectangle
- Draw height AX perpendicular to base CD
- Cut triangle △AXD from the left
- Move it to the right side
- The parallelogram becomes a rectangle of equal area
- Works because △AXD ≅ △BYC (by RHS congruency)
Rhombus
💎 Key Properties of Rhombus
- All sides are equal
- Diagonals are perpendicular bisectors of each other
- △ABD and △CBD are both isosceles triangles
- Each half can be converted to a rectangle, then joined → single rectangle of equal area
Rectangle WXYZ formed from rhombus ABCD dissection has:
Trapezium
⏢ What is a Trapezium?
- A quadrilateral with one pair of parallel sides
- The two parallel sides are called the parallel sides (a and b)
- The perpendicular distance between them is the height (h)
- Break it into 1 rectangle + 2 triangles to find area
Top parallel side
Bottom parallel side (longer)
Height (perpendicular distance between parallel sides)
⭐ Chapter Summary — All Formulas
- Area of a Rectangle = length × width
- Area of a Square = side²
- Area of a Triangle = ½ × base × height (holds for ALL triangles — acute, right, obtuse)
- Area of any Polygon = divide into triangles and add their areas
- Area of a Parallelogram = base × height (any side can be taken as base)
- Area of a Rhombus = ½ × d₁ × d₂ (product of diagonals)
- Area of a Trapezium = ½ × h × (a + b) (h = height, a & b = parallel sides)
- Perimeter ≠ Area — regions with the same perimeter can have different areas!
- Triangles between parallel lines with the same base always have the same area
- Dissection = cutting a shape into pieces and rearranging to form a different shape of equal area