🧩 Compound Area

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

  • Area formulae for triangles, quadrilaterals, and trapezoids
  • Split compound shapes into simpler parts
  • Find missing lengths using area and perimeter
  • Calculate shaded areas (subtraction method)
  • Solve real-life problems involving area

📖 Learn

1. Area Formulae Reference

ShapeFormulaNotes
RectangleA = l × wlength × width
TriangleA = ½ × b × hh = perpendicular height
ParallelogramA = b × hh = perpendicular height (not slant)
TrapezoidA = ½(a+b)×ha, b = parallel sides; h = perp. height
CircleA = πr²r = radius
KiteA = ½ × d₁ × d₂d₁, d₂ = diagonals
💡 h is ALWAYS the perpendicular height — the distance measured at right angles to the base.

2. Splitting Compound Shapes

Break any compound shape into rectangles, triangles, etc. then ADD the parts.

L-shape: split into 2 rectangles (horizontal or vertical cut)
T-shape: split into 3 rectangles or 2
Arrow/cross: identify the rectangular parts
Key: find any missing lengths first by subtraction
💡 Always label the missing dimensions using the given lengths before calculating.

3. Shaded Areas (Subtraction)

For a shaded region: Shaded area = Large area − Small area (cutout)

Square with hole: Area = square − circle
Frame: Area = outer rectangle − inner rectangle
Ring: Area = π(R² − r²)
⚠️ Make sure you subtract the correct shape. Identify what is NOT shaded.

4. Missing Lengths from Area

If you know the area, you can find a missing dimension by rearranging.

Rectangle: A = lw → l = A ÷ w
Triangle: A = ½bh → h = 2A ÷ b
Trapezoid: A = ½(a+b)h → can solve for h, a or b
Example: Triangle area = 24, base = 8 → h = 2×24÷8 = 6 cm

5. Units and Accuracy

cm² → m²: divide by 10 000 (1 m = 100 cm, so 1 m² = 10 000 cm²)
m² → cm²: multiply by 10 000
mm² → cm²: divide by 100
Significant figures: give answers to 3 s.f. unless told otherwise
💡 Always state units in your answer (cm², m², etc.)

✏️ Worked Examples

Example 1 – L-shape

L-shape: outer dimensions 10×8, notch cut out of top-right 4×5. Find area.

Method 1 (add): Split into two rectangles. Bottom: 10×3=30. Left column: 6×8=48. Total = 78 cm²
Method 2 (subtract): Full rectangle 10×8=80. Minus notch 4×5=20. Area = 80−20 = 60 cm²
Wait — let's recalculate with consistent dimensions: outer 10×8, cutout 4×5 from top-right → 80−20 = 60 cm²

Example 2 – Trapezoid

Trapezoid: parallel sides 6 cm and 10 cm, perpendicular height 5 cm. Find area.

A = ½(a+b)×h = ½(6+10)×5 = ½×16×5 = 40 cm²

Example 3 – Shaded Region

Square 12 cm side with a triangle cut from one corner (base 5 cm, height 4 cm). Find shaded area.

Square: 12×12 = 144 cm²
Triangle: ½×5×4 = 10 cm²
Shaded: 144 − 10 = 134 cm²

Example 4 – Missing Length

A trapezoid has area 60 cm², height 6 cm, one parallel side 8 cm. Find the other parallel side.

A = ½(a+b)×h: 60 = ½(8+b)×6 = 3(8+b)
8+b = 20: b = 12 cm

🎨 Visualizer

🧩 Shape Area Calculator

Select a shape and enter dimensions.

➖ Shaded Area Calculator

Rectangle minus a smaller rectangle inside it.

Exercise 1 – Basic Area Formulae

Calculate each area. Give exact or 2 d.p. answers.

Exercise 2 – Trapezoid Area

A = ½(a+b)×h

Exercise 3 – Compound Shapes (Addition)

Split into parts and add.

Exercise 4 – Shaded Areas (Subtraction)

Exercise 5 – Finding Missing Lengths from Area

📝 Practice – 20 Questions

🏆 Challenge – 8 Questions