Area of Plain Figure

UTKALPRESS

AREA OF PLANE FIGURES — ARITHMETIC HANDBOOK

PAGE 1 — BASIC CONCEPT OF AREA

Area is the amount of surface enclosed by a closed plane figure. It tells us how much two-dimensional space a figure occupies.

Area = Surface covered by a plane figure
Standard unit: square unit such as cm², m², km²

1. Important Units

Unit	Equivalent
1 m²	10,000 cm²
1 km²	10,00,000 m²
1 hectare	10,000 m²
1 acre	43,560 ft² approximately

2. Rectangle

A rectangle has four right angles. Its opposite sides are equal.

Area of Rectangle = Length × Breadth
A = l × b

Perimeter = 2(l + b)

Example: A rectangular field is 25 m long and 16 m wide. Find its area.

Area = 25 × 16 = 400 m²

3. Square

A square has four equal sides and four right angles.

Area of Square = Side × Side = a²
Perimeter = 4a
Diagonal = a√2

Example: Find the area of a square whose side is 12 cm.

Area = 12² = 144 cm²

4. Triangle

Area of Triangle = ½ × Base × Height
A = ½bh

Example: Base = 18 cm and height = 10 cm.
Area = ½ × 18 × 10 = 90 cm²

Remember: The height of a triangle must be perpendicular to the chosen base.

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AREA OF PLANE FIGURES

PAGE 2 — QUADRILATERALS & CIRCLE

1. Parallelogram

A parallelogram has opposite sides parallel and equal.

Area = Base × Corresponding Height
A = bh
Perimeter = 2(a + b)

Example: A parallelogram has base 20 cm and height 9 cm.
Area = 20 × 9 = 180 cm²

2. Rhombus

A rhombus has four equal sides. Its diagonals intersect at right angles and bisect each other.

Area of Rhombus = ½ × d₁ × d₂
Perimeter = 4a

Example: The diagonals of a rhombus are 16 cm and 12 cm.
Area = ½ × 16 × 12 = 96 cm²

3. Trapezium

A trapezium has one pair of parallel sides. Let the parallel sides be a and b and height be h.

Area of Trapezium = ½(a + b)h

Example: Parallel sides are 18 cm and 12 cm and height is 8 cm.
Area = ½(18 + 12) × 8
= ½ × 30 × 8
= 120 cm²

4. Circle

A circle is a plane figure in which every point on the circumference is at the same distance from the centre.

Area of Circle = πr²
Circumference = 2πr
Diameter = 2r
Therefore, r = d/2

Usually, use π = 22/7 or π = 3.14159, depending on the question.

Example: Find the area of a circle with radius 7 cm.

Area = 22/7 × 7 × 7
= 154 cm²

5. Semicircle

Area of Semicircle = ½πr²
Curved Length = πr
Total Perimeter = πr + 2r

Exam Tip: Do not confuse the curved length of a semicircle with its complete perimeter. The diameter must be added for total perimeter.

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AREA OF PLANE FIGURES

PAGE 3 — COMPOUND FIGURES & SPECIAL CASES

1. Sector of a Circle

A sector is the portion of a circle enclosed by two radii and an arc. If the central angle is θ degrees:

Area of Sector = (θ/360) × πr²
Arc Length = (θ/360) × 2πr

Example: Find the area of a sector of angle 90° and radius 14 cm.

Area = 90/360 × 22/7 × 14²
= ¼ × 22/7 × 196
= 154 cm²

2. Quadrant

A quadrant is one-fourth of a circle.

Area of Quadrant = ¼πr²
Arc Length = ½πr

3. Annulus / Ring

A ring-shaped region between two concentric circles is called an annulus. Let outer radius = R and inner radius = r.

Area of Ring = π(R² − r²)

Example: Outer radius = 10 cm and inner radius = 6 cm.
Area = π(10² − 6²)
= π(100 − 36)
= 64π cm²

4. Area of a Path

When a path surrounds a rectangular field, the area of the path is:

Area of Path = Area of Outer Rectangle − Area of Inner Rectangle

Example: A rectangular garden is 40 m × 30 m. A path surrounds it externally with width 2 m.

Outer dimensions = 44 m × 34 m
Outer area = 44 × 34 = 1496 m²
Inner area = 40 × 30 = 1200 m²
Area of path = 1496 − 1200 = 296 m²

5. Area of Four Walls / Flooring

Area of Four Walls = 2h(l + b)

Area of Floor = l × b

Example: A room is 8 m long, 6 m broad and 4 m high. Find the area of its four walls.

Area = 2 × 4 × (8 + 6)
= 8 × 14
= 112 m²

6. Equilateral Triangle

Area = (√3/4)a²
Height = (√3/2)a
Perimeter = 3a

Example: Find the area of an equilateral triangle of side 10 cm.
Area = √3/4 × 10²
= 25√3 cm²

Shortcut: For complicated figures, divide the figure into rectangles, triangles, squares, circles or semicircles. Find each area separately and then add or subtract.

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FORMULA SHEET • SOLVED EXAMPLES • PRACTICE

PAGE 4 — COMPLETE FORMULA REVISION

Figure	Area Formula
Square	a²
Rectangle	l × b
Triangle	½bh
Parallelogram	bh
Rhombus	½d₁d₂
Trapezium	½(a+b)h
Circle	πr²
Semicircle	½πr²
Sector	(θ/360)πr²
Annulus	π(R²−r²)
Equilateral Triangle	(√3/4)a²

SOLVED REVISION EXAMPLES

Example 1: A rectangle has area 360 m² and length 24 m. Find its breadth.

Breadth = Area ÷ Length
= 360 ÷ 24
= 15 m

Example 2: Find the radius of a circle whose area is 154 cm².

πr² = 154
22/7 × r² = 154
r² = 49
r = 7 cm

Example 3: A triangle has area 120 cm² and base 15 cm. Find its height.

½ × 15 × h = 120
15h = 240
h = 16 cm

PRACTICE QUESTIONS

1. Find the area of a square of side 18 cm.

2. Find the area of a rectangle measuring 35 m × 12 m.

3. Find the area of a triangle with base 24 cm and height 15 cm.

4. Find the area of a parallelogram with base 18 cm and height 11 cm.

5. Find the area of a rhombus whose diagonals are 20 cm and 14 cm.

6. Find the area of a circle with radius 14 cm. Use π = 22/7.

7. Find the area of a trapezium whose parallel sides are 25 cm and 15 cm and height is 10 cm.

8. Find the area of a semicircle of radius 14 cm. Use π = 22/7.

9. A room is 10 m long, 8 m broad and 4 m high. Find the area of its four walls.

10. A garden is 30 m × 20 m. A path of width 2 m is constructed outside it. Find the area of the path.

ANSWER KEY

1. 324 cm²
2. 420 m²
3. 180 cm²
4. 198 cm²
5. 140 cm²
6. 616 cm²
7. 200 cm²
8. 308 cm²
9. 144 m²
10. 216 m²

EXAM SHORTCUTS

1. Always convert all measurements into the same unit before calculating area.

2. Area is expressed in square units, not ordinary units.

3. For a composite figure, split it into familiar shapes.

4. For a path, calculate outer area − inner area.

5. For a circle, check whether the given measurement is radius or diameter before applying πr².

6. If diameter is given, use r = d/2.