Area · Lesson 12 of 12
Chapter Summary and Practice
“Consolidate the chapter through formula reasoning, common-error checks, mixed worked examples, conceptual questions, and cumulative practice.”
• Connect all major area formulas to the geometric ideas from which they were derived. • Choose an efficient strategy for rectangles, triangles, polygons, parallelograms, rhombi, trapeziums, and compound regions. • Distinguish area from perimeter and identify correct base-height relationships. • Convert square units correctly and reason about scale factors. • Solve mixed multi-step problems and explain the reasoning behind each method.
This chapter is not a list of unrelated formulas. Nearly every result grew from a small set of ideas: count square units, split a shape into simpler parts, rearrange pieces without changing area, use a perpendicular height, and compare regions that share a base or height.
Use this lesson as a complete revision. Work through the formula connections first, then the examples, quizzes, and mixed practice. When solving a new problem, identify the geometry before choosing arithmetic.
The Big Ideas
Area measures enclosed surface, while perimeter measures boundary length. Equal perimeter does not guarantee equal area, and equal area does not guarantee congruence or equal perimeter.
Triangles are central because polygons can be triangulated. Parallelogram, rhombus, and trapezium formulas can all be derived by dissection or triangle reasoning rather than memorised in isolation.
Formula Map
| Shape or situation | Area relationship |
|---|---|
| Rectangle | A = lw |
| Square | A = s² |
| Triangle | A = ½bh |
| Parallelogram | A = bh |
| Rhombus | A = bh or ½d₁d₂ |
| Trapezium | A = ½h(a + b) |
| Polygon | Split into non-overlapping triangles or simpler regions |
| Outer region with a hole/path | Outer area − inner area |
In every formula containing h, height means perpendicular height. The base-height pair must correspond. In the trapezium formula, a and b are specifically the parallel sides.
Common Errors to Catch
Check both the meaning and unit. Perimeter uses cm, m, and other length units; area uses cm², m², and other square units.
A sloping side of a triangle, parallelogram, or trapezium is not automatically the height.
Triangle, diagonal-rhombus, and trapezium formulas each contain a factor ½ for a geometric reason.
When adding overlapping strips or regions, subtract the overlap that was counted more than once.
If 1 ft = 12 in, then 1 ft² = 12² = 144 in².
Mixed Worked Examples
Problem
A triangle has area 70 cm² and base 20 cm. Find its height.
- 1.Use 70 = ½ × 20 × h.
- 2.70 = 10h.
- 3.h = 7 cm.
Problem
A rectangle and a parallelogram both have base 12 cm. The rectangle height is 7 cm and the parallelogram height is 5 cm. Compare their areas.
- 1.Rectangle area = 12 × 7 = 84 cm².
- 2.Parallelogram area = 12 × 5 = 60 cm².
- 3.The rectangle covers 24 cm² more because its perpendicular height is larger.
Problem
A rhombus has diagonals 26 cm and 10 cm. Find its area.
- 1.A = ½d₁d₂.
- 2.A = ½ × 26 × 10.
- 3.A = 130 cm².
Problem
A trapezium has area 120 cm², height 8 cm, and one parallel side 11 cm. Find the other parallel side.
- 1.120 = ½ × 8 × (11 + b).
- 2.120 = 4(11 + b), so 30 = 11 + b.
- 3.b = 19 cm.
Problem
Two 2 m wide paths cross through a 20 m by 15 m rectangular plot, one along the full length and one along the full width. Find total path area.
- 1.First strip: 20 × 2 = 40 m².
- 2.Second strip: 15 × 2 = 30 m².
- 3.Overlap: 2 × 2 = 4 m².
- 4.Total = 40 + 30 − 4 = 66 m².
Problem
Convert 2 ft² to in².
- 1.1 ft = 12 in, so 1 ft² = 144 in².
- 2.2 ft² = 2 × 144.
- 3.2 ft² = 288 in².
Strategy Checklist
Before calculating, ask: What is the whole region? Can it be split or rearranged? Which sides are parallel? Which segment is actually perpendicular? Is there a common base or common height? Would subtraction be shorter than addition? Are the units compatible? These questions often matter more than the arithmetic.
Choose triangle, parallelogram, rhombus, or trapezium. Without starting from its formula, draw a dissection or enclosing shape that leads to the formula. Write a short explanation of what stays unchanged and where each factor in the final expression comes from.
Concept Check
Quiz
Which statement is always true?
Which expression gives triangle area?
A parallelogram has base 15 cm and height 4 cm. Its area is
A rhombus has diagonals 12 cm and 18 cm. Its area is
A trapezium has parallel sides 8 cm and 14 cm and height 6 cm. Its area is
Triangles on the same base between the same parallels have
A square's side is tripled. Its area is multiplied by
If 1 m = 100 cm, then 1 m² equals
Cumulative Practice
Practice Problems
- A rectangle measures 18 cm by 7 cm. Find its area and perimeter, and explain why the units differ.
- Give two rectangles with the same perimeter but different areas.
- A triangle has area 90 cm² and height 12 cm. Find its base.
- A triangle has one base-height pair 15 cm and 8 cm. A second base is 10 cm. Find the corresponding second height.
- A 16 cm by 9 cm rectangle has both diagonals drawn. Find the area of each of the four triangles.
- A triangle has area 68 cm². A median is drawn. Find the area of each part.
- Parallelograms share a 13 cm base between parallel lines 7 cm apart. Find the area of each and state whether their perimeters must be equal.
- A parallelogram has area 96 cm² and base 12 cm. Find its height.
- A rhombus has area 180 cm² and one diagonal 20 cm. Find the other diagonal.
- A trapezium has parallel sides 15 cm and 27 cm and height 8 cm. Find its area.
- A trapezium has area 210 cm², height 10 cm, and one parallel side 16 cm. Find the other parallel side.
- A 24 m by 18 m park has a 2 m wide path outside it on every side. Find the path area.
- Two 3 m wide paths cross through a 30 m by 20 m field. Find the total area of the paths.
- Convert 50 in² to cm² using 1 in² = 6.4516 cm².
- Convert 1.2 km² to m².
- Explain with a drawing why two identical trapeziums can form a parallelogram.
- Describe a reflection construction for the shortest route from one point to another while touching a straight boundary.
- Explain why doubling every length of a figure multiplies its area by four.
Key Takeaways
• Area measures surface in square units; perimeter measures boundary in length units. • Triangle area is ½bh, and base-height reasoning drives many equal-area results. • Any polygon can be handled by decomposition into non-overlapping simpler regions. • Parallelogram area is bh; rhombus area can also be written ½d₁d₂. • Trapezium area is ½h(a + b), with a and b the parallel sides. • Dissection and reflection are reasoning tools, not separate formula lists. • Scaling all lengths by k scales area by k². • Area-unit conversions square the corresponding length conversion factor.
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Areas in Real Life
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