Area and Perimeter
~13 min read
- What: Areas and perimeters of squares, rectangles, triangles, parallelograms, rhombuses, trapeziums, circles, and sectors. Heron's formula for general triangles.
- Why it matters: CDS averages 4–6 questions per paper on mensuration. Formulas are short and answer-clean.
- Key fact: For a triangle with sides \(a, b, c\) and semi-perimeter \(s = (a + b + c)/2\), Heron's formula gives area \(= \sqrt{s(s - a)(s - b)(s - c)}\). Works for any triangle without needing a height.
Area and Perimeter is one of those chapters where pure formula recall wins the marks. CDS recycles the same handful of shapes and the same set of "find missing dimension" archetypes — once you have the dozen formulas locked, the chapter delivers 4–6 marks per paper at high pace.
This page is built from CDS Previous Year Questions across 2000–2023, plus NCERT Class 8 Area, Class 9 Perimeter and Area, and Class 10 Areas Related to Circles. Pair with Volume and Surface Area (3D extension) and Triangles and Properties.
What This Topic Covers
CDS scope: (1) basic shapes — square, rectangle, triangle, parallelogram, rhombus, trapezium; (2) circle — circumference, area, sector area, arc length; (3) Heron's formula for general triangles; (4) composite figures — combining basic shapes; (5) inscribed and circumscribed figures; and (6) percentage change in area when dimensions change.
Why This Topic Matters
- 4–6 CDS questions per paper, with clean numerical answers.
- Formula-heavy chapter — memorisation pays off massively.
- Percentage-change-in-area problems combine with Percentage for two-chapter cross-training.
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2007-I/II | 4 | Rectangle, triangle, sector |
| 2008-I/II | 4 | Circle, trapezium |
| 2010-I/II | 5 | Heron, square, percentage change |
| 2011-I/II | 4 | Composite, rhombus |
| 2012-I/II | 5 | Mixed |
| 2013-I/II | 5 | Sector, triangle area |
| 2014-I/II | 4 | Circle inscribed in square |
| 2015-I/II | 5 | Mixed |
| 2016-I/II | 5 | Trapezium, parallelogram |
| 2017-I/II | 5 | Heron, percentage change |
| 2018-I/II | 4 | Composite |
| 2019-II | 3 | Mixed |
| 2020-I/II | 4 | Sector, segment |
| 2021-I/II | 4 | Mixed |
| 2022-I / 2023-I | 5 | Mixed |
If a rectangle's length is increased by \(x\%\) and breadth by \(y\%\), the new area changes by \(x + y + xy/100\,\%\) — the successive-percentage formula applied to two dimensions. Memorise this one-liner.
Core Concepts
Square and Rectangle
Triangle
Equilateral with side \(a\): Area \(= \tfrac{\sqrt{3}}{4} a^2\); Height \(= \tfrac{\sqrt{3}}{2} a\).
Right triangle with legs \(p, q\): Area \(= \tfrac{1}{2} pq\).
Heron's formula: \(s = (a + b + c)/2\); Area \(= \sqrt{s(s-a)(s-b)(s-c)}\).
Parallelogram, Rhombus, Trapezium
Rhombus: Area \(= \tfrac{1}{2} d_1 d_2\) (product of diagonals over 2).
Trapezium: Area \(= \tfrac{1}{2}(a + b) h\) where \(a, b\) are parallel sides and \(h\) is the perpendicular distance.
Circle and Sector
Sector with angle \(\theta\) (degrees): Arc length \(= \tfrac{\theta}{360} \cdot 2\pi r\); Area \(= \tfrac{\theta}{360} \cdot \pi r^2\).
Segment area = sector area − triangle area.
Percentage Change in Area
Square: side \(+x\%\) → area \(+ (2x + x^2/100)\%\).
Circle: radius \(+x\%\) → area \(+ (2x + x^2/100)\%\).
Worked Examples
Example 1 — Triangle Area (2010-I)
Q: Find the area of a triangle with sides 3, 4, 5.
- Recognise 3-4-5 as a right triangle.
- Area = \(\tfrac{1}{2} \cdot 3 \cdot 4 = 6\) sq units.
Example 2 — Heron's Formula (2017-I)
Q: Find the area of a triangle with sides 13, 14, 15.
- \(s = (13 + 14 + 15)/2 = 21\).
- Area \(= \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84\) sq units.
Example 3 — Equilateral Triangle (2014-II)
Q: Find the area of an equilateral triangle with side 8 cm.
- Area \(= \tfrac{\sqrt{3}}{4} \cdot 8^2 = 16\sqrt{3} \approx 27.7\) sq cm.
Example 4 — Sector (2013-II)
Q: Find the area of a sector with radius 14 cm and central angle 60°.
- Area \(= \tfrac{60}{360} \cdot \pi \cdot 14^2 = \tfrac{1}{6} \cdot \pi \cdot 196 = \tfrac{196\pi}{6} \approx 102.6\) sq cm.
Example 5 — Percentage Change (2016-II)
Q: The length of a rectangle is increased by 20% and breadth decreased by 10%. Find the percentage change in area.
- Apply successive: \(20 - 10 + (20)(-10)/100 = 10 - 2 = 8\%\) increase.
Example 6 — Rhombus (2018-I)
Q: The diagonals of a rhombus are 24 cm and 10 cm. Find its area.
- Area \(= \tfrac{1}{2} d_1 d_2 = \tfrac{1}{2} \cdot 24 \cdot 10 = 120\) sq cm.
Example 7 — Trapezium (2015-I)
Q: The parallel sides of a trapezium are 9 cm and 7 cm, and its height is 5 cm. Find its area.
- Area \(= \tfrac{1}{2}(9 + 7) \cdot 5 = \tfrac{1}{2} \cdot 16 \cdot 5 = 40\) sq cm.
How CDS Tests This Topic
Six recurring archetypes: (1) basic shape area or perimeter given dimensions, (2) Heron's formula for general triangle, (3) sector or segment of a circle, (4) percentage change in area when dimensions change, (5) composite figure (e.g. semicircle on top of rectangle), (6) inscribed or circumscribed (square in circle, circle in square).
Exam Shortcuts (Pro-Tips)
Shortcut 1 — Memorise Heron-Friendly Triangles
13-14-15 → area 84; 9-10-17 → area 36; 5-5-6 → area 12; 5-5-8 → area 12. Common CDS triangles with clean Heron answers.
Shortcut 2 — Equilateral Quick
Equilateral with side \(a\): area \(= \tfrac{\sqrt{3}}{4} a^2\). For \(a = 4\): area \(= 4\sqrt{3}\); \(a = 6\): area \(= 9\sqrt{3}\); \(a = 8\): area \(= 16\sqrt{3}\). Recognise these.
Shortcut 3 — Diagonal Identities
Square side \(a\): diagonal \(a\sqrt{2}\). Rectangle \(\ell \times b\): diagonal \(\sqrt{\ell^2 + b^2}\). Rhombus: diagonals are perpendicular bisectors of each other.
Shortcut 4 — Sector Identity
Sector area / circle area = \(\theta/360\). Use this ratio directly when computing.
Shortcut 5 — Percentage on Area
Single-dimension: \(x\% \to 2x + x^2/100\%\) change in area. Two-dimension: \(x + y + xy/100\%\).
Common Question Patterns
Pattern 1 — Find Area from Dimensions
Standard formula application. Rectangle, triangle, parallelogram, etc.
Pattern 2 — Heron for General Triangle
Three sides given, no height. Use \(s = (a + b + c)/2\) then \(\sqrt{s(s-a)(s-b)(s-c)}\).
Pattern 3 — Sector or Segment
Circle with given central angle. Apply \(\theta/360\) ratio.
Pattern 4 — Percentage Change
Dimensions change by some percent. Apply two-dimension successive formula.
Pattern 5 — Composite Figure
Combine known shape areas. E.g. semicircle on a rectangle.
Preparation Strategy
Week 1. Memorise all formulas. Drill 20 single-shape problems. Practice Heron on common triangles (13-14-15, 9-10-17, etc.).
Week 2. Percentage change, sector and segment, composite figures. Cross-train with Percentage.
Mock testing. Use CDS mock tests. Common slip-ups: square-vs-rectangle confusion in area, forgetting \(\pi\) in circle, sign errors in percentage change. Drill until clean.
Drill Area and Perimeter
CDS mocks with all shape variants, Heron's formula, sectors, and percentage change. Six archetypes — twelve formulas — reflex.
Start Free Mock TestFrequently Asked Questions
What is Heron's formula?
For a triangle with sides \(a, b, c\) and semi-perimeter \(s = (a + b + c)/2\): area \(= \sqrt{s(s - a)(s - b)(s - c)}\). It works for any triangle without needing the height. Try it on the 3-4-5 triangle: \(s = 6\); area = \(\sqrt{6 \cdot 3 \cdot 2 \cdot 1} = 6\). ✓
What's the area of an equilateral triangle?
For side \(a\): area \(= \tfrac{\sqrt{3}}{4} a^2\). For \(a = 2\): area \(= \sqrt{3}\); for \(a = 4\): area \(= 4\sqrt{3}\); for \(a = 8\): area \(= 16\sqrt{3}\).
What's the area of a rhombus?
If the two diagonals have lengths \(d_1\) and \(d_2\): area \(= \tfrac{1}{2} d_1 d_2\). Example: diagonals 24 and 10 give area = 120.
How do I find the area of a sector?
Sector area \(= (\theta / 360°) \cdot \pi r^2\), where \(\theta\) is the central angle in degrees. The fraction \(\theta/360\) is the portion of the full circle the sector represents.
If the length of a rectangle increases by 20% and breadth decreases by 10%, what happens to the area?
Apply the two-dimension successive formula: \(20 + (-10) + (20)(-10)/100 = 10 - 2 = 8\%\) increase. Always sign-correct: increase positive, decrease negative.
What's the relationship between diagonal and side of a square?
For a square with side \(a\), the diagonal is \(a\sqrt{2}\). So a square with side 1 has diagonal \(\sqrt{2} \approx 1.414\). The diagonal of a rectangle is \(\sqrt{\ell^2 + b^2}\).
Which CDS Maths topics connect to Area and Perimeter?
Volume and Surface Area — the 3D extension. Triangles and Properties — formulas for triangle area. Circles — sector and segment areas. Percentage — dimension-change problems.