Area and Perimeter
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Important Formulas & Concepts
Study MaterialArea and Perimeter
Area and Perimeter are among the most important topics in Quantitative Aptitude and Mensuration. Questions from this chapter are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, Defence, and various competitive examinations.
This chapter deals with measuring:
- Boundary length of plane figures
- Space enclosed by figures
- Properties of triangles and quadrilaterals
- Circle measurements
- Mensuration formulas
- Geometry-based aptitude problems
Strong understanding of formulas, geometric properties, and calculation techniques helps candidates solve mensuration problems quickly and accurately.
What is Perimeter?
The total length of the boundary of a closed plane figure is called its perimeter.
Perimeter = Sum of all sides
Perimeter is measured in linear units like:
- cm
- m
- km
What is Area?
The total space enclosed inside a plane figure is called its area.
Area = Space enclosed within a figure
Area is measured in square units like:
- cm²
- m²
- km²
Important Geometry Concepts
1. Triangle
A triangle is a polygon having three sides and three angles.
Sum of angles of a triangle = 180°
2. Quadrilateral
A quadrilateral is a polygon having four sides.
Sum of angles of a quadrilateral = 360°
3. Circle
A circle is a set of all points at equal distance from a fixed point called the center.
Important terms:
- Radius
- Diameter
- Circumference
- Chord
- Arc
- Sector
Important Formulas for Triangle
| Concept | Formula |
|---|---|
| Perimeter | a + b + c |
| Area | 1/2 × Base × Height |
| Semi-perimeter | (a + b + c)/2 |
| Heron's Formula | √[s(s−a)(s−b)(s−c)] |
| Equilateral Triangle Area | (√3/4)a² |
| Equilateral Triangle Height | (√3/2)a |
| Right Triangle Formula | a² = b² + c² |
Important Properties of Triangle
- Sum of any two sides is greater than the third side.
- The median divides triangle into two equal areas.
- Centroid divides median in ratio 2:1.
- In equilateral triangle, all sides and angles are equal.
- In isosceles triangle, two sides are equal.
- In right triangle, one angle is 90°.
Types of Triangles
| Type | Property |
|---|---|
| Equilateral Triangle | All sides equal |
| Isosceles Triangle | Two sides equal |
| Scalene Triangle | All sides unequal |
| Right Triangle | One angle = 90° |
| Acute Triangle | All angles < 90° |
| Obtuse Triangle | One angle > 90° |
Important Formulas for Rectangle
| Concept | Formula |
|---|---|
| Perimeter | 2(l + b) |
| Area | l × b |
| Diagonal | √(l² + b²) |
Important Formulas for Square
| Concept | Formula |
|---|---|
| Perimeter | 4a |
| Area | a² |
| Diagonal | a√2 |
| Area using diagonal | d²/2 |
Important Formulas for Parallelogram
| Concept | Formula |
|---|---|
| Perimeter | 2(a + b) |
| Area | Base × Height |
Important properties:
- Opposite sides are equal and parallel.
- Diagonals bisect each other.
- Diagonal divides parallelogram into equal triangles.
Important Formulas for Rhombus
| Concept | Formula |
|---|---|
| Perimeter | 4a |
| Area | 1/2 × d₁ × d₂ |
Important properties:
- All sides are equal.
- Diagonals bisect at right angles.
Important Formulas for Trapezium
| Concept | Formula |
|---|---|
| Area | 1/2 × (Sum of parallel sides) × Height |
Important Formulas for Circle
| Concept | Formula |
|---|---|
| Circumference | 2πr |
| Area | πr² |
| Diameter | 2r |
| Arc Length | (θ/360) × 2πr |
| Sector Area | (θ/360) × πr² |
| Semi-circle Area | πr²/2 |
Important Circle Concepts
- Diameter is twice the radius.
- Longest chord of a circle is diameter.
- Angle subtended by diameter is 90°.
- Circumference means perimeter of circle.
Important Polygon Formulas
| Polygon | Formula |
|---|---|
| Interior Angle Sum | (n − 2) × 180° |
| Each Interior Angle of Regular Polygon | [(n−2) × 180]/n |
| Exterior Angle | 360°/n |
Mensuration Conversion Formulas
| Conversion | Value |
|---|---|
| 1 m² | 10,000 cm² |
| 1 hectare | 10,000 m² |
| 1 km² | 1,000,000 m² |
Important Geometry Properties
- Diagonals of rectangle are equal.
- Diagonals of square are equal and perpendicular.
- Area of square increases by square of side increase.
- Perimeter depends on linear dimensions.
- Area depends on square dimensions.
Error and Percentage Concepts
If side changes by x%:
Percentage change in area = 2x + x²/100
Applications of Area and Perimeter
- Construction measurements
- Land area calculations
- Road and park measurements
- Architecture and design
- Geometry aptitude exams
Common Mistakes in Area and Perimeter
- Using wrong formulas.
- Confusing perimeter and area.
- Ignoring unit conversions.
- Calculation mistakes in square roots.
- Incorrect use of π value.
Important Exam Tips
- Memorize all standard formulas.
- Practice geometry diagrams regularly.
- Use π = 22/7 when suitable.
- Draw figures for better understanding.
- Learn properties of triangles and quadrilaterals.
- Practice area comparison questions.
- Verify units carefully.
Quick Revision Formula Table
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle | 2(l+b) | l×b |
| Square | 4a | a² |
| Triangle | a+b+c | 1/2 × b × h |
| Circle | 2πr | πr² |
| Parallelogram | 2(a+b) | b×h |
| Rhombus | 4a | 1/2 × d₁ × d₂ |
Area and Perimeter form the foundation of Mensuration and Geometry. Strong understanding of formulas, geometric properties, and calculation techniques helps candidates solve aptitude problems quickly and accurately in competitive examinations.