Area
Area - Methods to calculate area of square, rectangle, triangle, circle and other shapes
Learning Content
📐 What is Area?
The surface of a two-dimensional figure or the space it occupiesSpread(Area) we say.
Spreadin square units(Square Units) are measured.
📏 Units
| unit of length | unit of area |
|---|---|
| meter (m) | square meter (m²) |
| Centimeter (cm) | Square cm. (cm²) |
| kilometer (km) | sq. km. (km²) |
| feet (ft) | square feet (sq.ft) |
🔄 Unit Conversions
| Change | value |
|---|---|
| 1 m² | 10000 cm² |
| 1 km² | 1000000 m² (10⁶ m²) |
| 1 hectare | 10000 m² |
| 1 acre | 4046.86 m² ≈ 4047 m² |
| 1 km² | 100 hectares |
| 1 sq.ft | 929.03 cm² ≈ 0.0929 m² |
📊 Basic shapes
1. Square
All four sides are equal in shape
If side = a, then area = a²
2. Rectangle
Opposite sides are congruent
If length = l, breadth = b, then area = l × b
3. Triangle
Three page format
If base = b, height = h, area = ½ × b × h
4. Circle
Points equidistant from the center
If radius = r, then area = πr²
📚 Key Codes
| Code | Meaning | English |
|---|---|---|
| a | side of the square | Side of Square |
| l | length | Length |
| b | Width / Bottom | Breadth / Base |
| h | height | Height |
| r | Radius | Radius |
| d | diameter | Diameter |
| π | Pi = 22/7 or 3.14 | Pi |
| P | Circumference | Perimeter |
| A | Spread | Area |
🎯 Perimeter vs Area
| Perimeter | Area |
|---|---|
| The length of the border of the shape | The space inside the shape |
| One Dimensional Scale (1D) | Two Dimensional Scale (2D) |
| Meter, cm units | Square meter, square cm. |
| to fence | Spread the floor |
💡 Things to remember
- π (pi)= 22/7 = 3.14159... (for round calculations)
- √2= 1.414 (diagonal of square)
- √3= 1.732 (equilateral triangle)
- Diameter = 2 × radius (d = 2r)
- Diagonal of square = a√2
- Diagonal of rectangle = √(l² + b²)
📐 Area Formulas - All Shapes
All Area Formulas for TNPSC Exam!
◼️ Square
| Spread | A = a² | a = page |
| Circumference | P = 4a | |
| Diagonal | d = a√2 | |
| side (from surface) | a = √A | |
| side (from circumference) | a = P/4 | |
| Area (diagonal) | A = d²/2 |
▬ Rectangle
| Spread | A = l × b | l = length, b = width |
| Circumference | P = 2(l + b) | |
| Diagonal | d = √(l² + b²) | |
| length | l = A/b | |
| width | b = A/l |
🔺 Triangle
| General formula | A = ½ × b × h | b = base, h = height |
| Circumference | P = a + b + c | a, b, c = sides |
| Heron formula | A = √[s(s-a)(s-b)(s-c)] | s = (a+b+c)/2 |
| Equilateral triangle | A = (√3/4) × a² | a = page |
| Isosceles triangle | A = ¼ × b × √(4a² - b²) | a = equilateral, b = foot |
| A right triangle | A = ½ × feet × height |
⭕ circle
| Spread | A = πr² | r = radius |
| Circumference | C = 2πr = πd | d = diameter |
| Spread in diameter | A = πd²/4 | |
| Radius (from surface) | r = √(A/π) | |
| Radius (from Circumference) | r = C/2π |
◗ Semicircle & Ring
| form | Spread | Circumference |
|---|---|---|
| semicircle | A = πr²/2 | P = πr + 2r = r(π + 2) |
| foot circle | A = πr²/4 | P = πr/2 + 2r |
| Ring | A = π(R² - r²) | R = outer radius, r = inner radius |
| Sector | A = (θ/360) × πr² | θ = central angle |
▰ Parallelogram
| Spread | A = b × h | b = base, h = height |
| Circumference | P = 2(a + b) | a, b = adjacent sides |
| Diagonal area | A = ½ × d₁ × d₂ × sinθ | d₁, d₂ = diagonals |
◇ Rhombus
| Spread | A = ½ × d₁ × d₂ | d₁, d₂ = diagonals |
| Circumference | P = 4a | a = page |
| page | a = ½ × √(d₁² + d₂²) |
⏢ Trapezium
| Spread | A = ½ × (a + b) × h | a, b = parallel sides, h = height |
| Circumference | P = a + b + c + d | A supplement of four pages |
⬡ Regular Polygons
| form | Spread |
|---|---|
| Equilateral triangle | A = (√3/4) × a² |
| A regular hexagon | A = (3√3/2) × a² |
| n-sided polygon | A = ¼ × n × a² × cot(π/n) |
📐 Pathway Formulas
Path area = 2w(l + b - 2w)
(w = track width, l = length, b = width)
Path area = 2w(l + b + 2w)
Path area = 4w(a + w) [out]
Path area = 4w(a - w) [in]
🔄 Key relationships
- If the diagonal d of the square is:Area = d²/2, side = d/√2
- In circle:Diameter d = 2r, perimeter = πd, area = πd²/4
- In an equilateral triangle:Height h = (√3/2) × a
- In the diagonal:a² = (d₁/2)² + (d₂/2)²
📝 Coverage - Solved Examples
10 Important Questions for TNPSC Exam
Question 1: Square
Question:The perimeter of a square is 64 cm. If so, what is its area?
Solution:
Perimeter P = 64 cm.
Side a = P/4 = 64/4 =16 cm.
Area A = a² = 16² =256 sq. cm.
Question 2: Rectangle
Question:Length of a rectangle is 15 m and width is 8 m. If , find its area and diagonal.
Solution:
Length l = 15 m, width b = 8 m.
Area A = l × b = 15 × 8 =120 sq m.
Diagonal d = √(l² + b²) = √(225 + 64) = √289 =17 m.
Question 3: Triangle
Question:The base of a triangle is 24 cm and the height is 15 cm. If so, find the area.
Solution:
Base b = 24 cm, height h = 15 cm.
Area A = ½ × b × h
A = ½ × 24 × 15 =180 sq. cm.
Question 4: Equilateral Triangle
Question:The side of an equilateral triangle is 14 cm. If so, find the area.
Solution:
Side a = 14 cm.
Area A = (√3/4) × a²
A = (√3/4) × 14² = (√3/4) × 196
A = 49√3 = 49 × 1.732 =84.87 sq. cm.
Question 5: Circle
Question:The radius of the circle is 14 cm. If so, find area and perimeter. (π = 22/7)
Solution:
Radius r = 14 cm.
Area A = πr² = (22/7) × 14 × 14
A = 22 × 2 × 14 =616 sq. cm.
Circumference C = 2πr = 2 × (22/7) × 14 =88 cm.
Question 6: Ring
Question:The outer radius of the ring is 21 cm and the inner radius is 14 cm. If so, what is the area of the ring?
Solution:
Outer radius R = 21 cm, Inner radius r = 14 cm.
Area of ring = π(R² - r²)
= (22/7) × (21² - 14²)
= (22/7) × (441 - 196)
= (22/7) × 245 = 22 × 35 =770 sq. cm.
Question 7: Rhombus
Question:The diagonals of a rhombus are 16 cm. and 12 cm. If so, see spread and page.
Solution:
d₁ = 16 cm, d₂ = 12 cm.
Area A = ½ × d₁ × d₂ = ½ × 16 × 12 =96 sq. cm.
Side a = ½ × √(d₁² + d₂²) = ½ × √(256 + 144)
a = ½ × √400 = ½ × 20 =10 cm
Question 8: Trapezium
Question:Parallel sides of a parallelogram are 20 cm. and 30 cm, height 15 cm. If so, find the area.
Solution:
Parallel sides a = 20 cm, b = 30 cm, height h = 15 cm.
Area A = ½ × (a + b) × h
A = ½ × (20 + 30) × 15
A = ½ × 50 × 15 =375 sq. cm.
Question 9: Heron's Formula
Question:The sides of a triangle are 13 cm, 14 cm, 15 cm. If so, find the area.
Solution:
Sides a = 13, b = 14, c = 15
Semicircumference s = (a + b + c)/2 = (13 + 14 + 15)/2 = 42/2 =21
s - a = 21 - 13 = 8
s - b = 21 - 14 = 7
s - c = 21 - 15 = 6
Area A = √[s(s-a)(s-b)(s-c)] = √(21 × 8 × 7 × 6)
A = √7056 =84 sq. cm.
Question 10: Pathway
Question:50 m. Length is 40 m. 2.5 m on the outside of a rectangular ground of width. There is a wide path. View the area of the track.
Solution:
Length l = 50 m., Width b = 40 m., Track width w = 2.5 m.
Method 1: Direct formula
Outer track area = 2w(l + b + 2w)
= 2 × 2.5 × (50 + 40 + 5)
= 5 × 95 =475 sq m.
Method 2: Differentiation
Outer rectangle = (50 + 5) × (40 + 5) = 55 × 45 = 2475 sq m.
Inner rectangle = 50 × 40 = 2000 sq m.
Path = 2475 - 2000 =475 sq m.
📚 Additional practice questions
- Diagonal of a square is 10√2 cm. If, area =100 sq. cm.
- The circumference of a circle is 44 cm. If, area =154 sq. cm.
- The area of the rectangle is 84 sq. m., width is 7 m. If, length =12 m.
- The base of the joint is 18 cm and the height is 12 cm. If, area =216 sq. cm.
- The radius of the semicircle is 7 cm. If, area =77 sq. cm.
⚡ Spread - cross paths
Super Tricks to Save Time in TNPSC Exam!
🚀 Trick 1: Square - Quick calculation
If diagonal d is:Area = d²/2
Example:Circumference = 40 cm.
Area = 40²/16 = 1600/16 =100 sq. cm.
🚀 Trick 2: Rectangle - Diagonal Shortcut
If sides are 3k, 4k, diagonal = 5k
5:12:13 aspect ratio rectangle:
If sides are 5k, 12k, diagonal = 13k
Example:Length = 12 m., Width = 9 m.
This is a 3:4 ratio (9:12 = 3:4), k = 3
Diagonal = 5 × 3 =15 m.
🚀 Trick 3: Circle - Key Values
| Radius (r) | Perimeter (2πr) | Area (πr²) |
|---|---|---|
| 7 | 44 | 154 |
| 14 | 88 | 616 |
| 21 | 132 | 1386 |
| 28 | 176 | 2464 |
| 35 | 220 | 3850 |
🚀 Trick 4: Page Change → Surface Change
Rectangle:Length x%, width y% increase → Area (x + y + xy/100)% increase
Example:10% increase on the side of the square
Increase in area = 2(10) + 100/100 = 20 + 1 =21%
🚀 Trick 5: Radius/Diameter conversion
Circumference → x% increase
Area → (2x + x²/100)% increase
Example:Doubling the radius (100% increase)
Circumference = 100% = double
Area = 2(100) + 10000/100 = 200 + 100 = 300% =Fourfold
🚀 Trick 6: Triangular memory
√3/2 = 0.866(equilateral height)
| Equilateral side (a) | Area = (√3/4)a² |
|---|---|
| 2 | √3 = 1.732 |
| 4 | 4√3 = 6.93 |
| 6 | 9√3 = 15.59 |
| 10 | 25√3 = 43.3 |
🚀 Trick 7: Pythagorean triples
| trinity | application |
|---|---|
| 3, 4, 5 | Basic (folds: 6,8,10; 9,12,15; 12,16,20) |
| 5, 12, 13 | Multiples: 10,24,26; 15,36,39 |
| 8, 15, 17 | Multiples: 16,30,34 |
| 7, 24, 25 | Multiples: 14,48,50 |
🚀 Trick 8: Slider - Quick method
= [(a + b)/2] × h
Example:Parallel sides are 16 cm, 24 cm, height 10 cm.
Average width = (16 + 24)/2 = 20 cm.
Area = 20 × 10 =200 sq. cm.
🚀 Trick 9: Route Quick Formulas
| Type | formula |
|---|---|
| Path out of the rectangle | 2w(l + b + 2w) |
| Path inside the rectangle | 2w(l + b - 2w) |
| Path out of the square | 4w(a + w) |
| Path inside the square | 4w(a - w) |
| Cross paths (+) | w(l + b - w) |
🚀 Trick 10: Heron Formula Shortcut
(s-a), (s-b), (s-c) Calculate → Multiply → √ Take
Example:Pages 5, 6, 7
s = (5+6+7)/2 = 9
s-a=4, s-b=3, s-c=2
Area = √(9×4×3×2) = √216 = 6√6 ≈14.7 square units
📊 TNPSC Frequently Asked Categories
| Question type | Crossroads |
|---|---|
| Spread from the perimeter | Square: P²/16, Circle: C²/4π |
| Spread across the diagonal | Square: d²/2, Diagonal: d₁d₂/2 |
| % change | x% → (2x + x²/100)% area |
| Circumference → Area | A = C²/4π = Cr/2 |
| Path width | Outer: 2w(l+b+2w), Inner: 2w(l+b-2w) |
💡 Important memory tips
- π = 22/7- For circular calculations (if radius is 7 times)
- π = 3.14- In other cases
- √2 = 1.414- Square Diagonal
- √3 = 1.732- Equilateral triangle
- √3/4 = 0.433- Equilateral area coefficient
- Circle:2 times r → 4 times the area
- square:2 times a → 4 times the area
- 1 hectare = 10000 m²
General Studies
All Parts
General Science
Nature of matter, physics, mechanics, electricity, magnetism...
Current events
Collection of Recent Events, National Symbols – Details of S...
Geography
Topography – Natural Formations – Monsoon, Rainfall, Weather...
History and Culture of India
Indus Valley Civilization – Guptas, Delhi Sultans, Mughals a...
Constitution of India
Constitution of India – Preamble to Constitution – Main Obje...
Indian Economy
Nature of Indian Economy – Five Year Plans – A Maturity – Pl...
Indian National Movement
National Revival, Indian National Congress, BR Ambedkar, Bha...
History, Culture, Tradition and Socio-Political Movements of Tamil Nadu
Tamil Community History, Thirukkural, Various Reformers, Ref...
Development Administration in Tamil Nadu
Social justice and social harmony, education and welfare sys...
Competence and mental intelligence
Compression – Percent – Greater Common Factor – Greater Co...