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CBSE · Class 9 · Mathematics

Chapter 6: Perimeter and Area

6 concepts · 48 practice questions with answers · Free to read

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Perimeter of plane shapes

From daily life

A farmer needs wire to go once round a field. The wire length is the perimeter, the distance round the boundary.

Simple definition

Perimeter is the total length of the boundary. Rectangle: 2(l + b). Square: 4a. Triangle: sum of sides. Regular polygon: n × side.

Formula: Rectangle 2(l + b); square 4a; unit: cm, m

8585P = 2(8+5)= 26

In the figure: A rectangle 8 by 5 has perimeter 2(8 + 5) = 26 units.

Rule 1: Find the perimeter — Add all outer sides; use the formula for rectangles and squares.

  • Easy: Square side 9 cm: 36 cm.
  • Medium: Triangle 5, 12, 13: 30.
  • Exam-type: Regular hexagon, side 4: 24.

Rule 2: Work backwards — Divide the perimeter by the number of equal sides, or use P = 2(l + b) to find a side.

  • Easy: Square perimeter 48: side 12.
  • Medium: Rectangle P = 30, l = 9: b = 6.
  • Exam-type: Fence 100 m around a square field: side 25 m.
Remember

Perimeter is a length (cm, m), not an area.

Common mistake

Using only l + b for a rectangle and forgetting to double.

Memory clue

"Walk all the way round."

Practice questions

Easy
  1. 1A rectangle is 28 cm long and 14 cm wide. Find its perimeter.
  2. 2A rectangle is 20 cm long and 18 cm wide. Find its perimeter.
  3. 3The perimeter of a square is 40 cm. What is the length of its side?
Medium
  1. 4The perimeter of a rectangle is 60 cm and its length is 15 cm. Find its width.
  2. 5The perimeter of a rectangle is 84 cm and its length is 21 cm. Find its width.
  3. 6A regular pentagon has each side 12 cm. Find its perimeter.
Challenging
  1. 7A regular octagon has each side 10 cm. Find its perimeter.
  2. 8The length of a rectangle is 8 cm more than twice its width, and its perimeter is 124 cm. Find its width.
Show answers (try them yourself first)
  1. 184 cm
  2. 276 cm
  3. 310 cm
  4. 415 cm
  5. 521 cm
  6. 660 cm
  7. 780 cm
  8. 818 cm

Circumference of a circle and π

From daily life

Roll a coin once along a ruler: it covers about 3.14 times its width. That ratio is the same for every circle.

Simple definition

The distance round a circle is its circumference C. The ratio C/d is the same for all circles and is called π (≈ 22/7 or 3.14). So C = πd = 2πr.

Formula: C = 2πr = πd; π ≈ 22/7 ≈ 3.14

dC = πd = 2πr

In the figure: With diameter d the circle has circumference πd, a little over 3 diameters.

Rule 1: Find C — Use 22/7 when the radius is a multiple of 7; otherwise 3.14.

  • Easy: r = 7: C = 2 × 22/7 × 7 = 44.
  • Medium: d = 10: C = 31.4 (π = 3.14).
  • Exam-type: r = 3.5: C = 22.

Rule 2: Find r from C or wheel turns — r = C/(2π). Distance = turns × circumference.

  • Easy: C = 88: r = 14.
  • Medium: Wheel C = 2 m, 50 turns: 100 m.
  • Exam-type: Wire of 44 cm bent into a circle: r = 7 cm.
Remember

π is irrational: 22/7 and 3.14 are only approximations.

Common mistake

Using the radius in C = πd without doubling it.

Memory clue

"Circle edge = pi times diameter."

Practice questions

Easy
  1. 1For every circle, the ratio of the circumference to the diameter is the same number, called
  2. 2The circumference of a circle of radius r is
  3. 3π is
Medium
  1. 4Find the circumference of a circle of radius 56 cm. (Take π = 22/7.)
  2. 5Find the circumference of a circle of radius 28 cm. (Take π = 22/7.)
  3. 6Find the circumference of a circle of radius 49 cm. (Take π = 22/7.)
Challenging
  1. 7Find the circumference of a circle of radius 14 cm. (Take π = 22/7.)
  2. 8Find the circumference of a circle of diameter 21 cm. (Take π = 22/7.)
Show answers (try them yourself first)
  1. 1π
  2. 22πr
  3. 3an irrational number, about 3.14159
  4. 4352 cm
  5. 5176 cm
  6. 6308 cm
  7. 788 cm
  8. 866 cm

Length of an arc

From daily life

A pizza slice has a curved crust. Its crust is a fraction of the full circle edge, the same fraction as the angle at the tip is of 360°.

Simple definition

An arc making angle θ at the centre has length (θ/360) × 2πr. Perimeter of a sector = arc + 2r.

Formula: l = (θ/360) × 2πr

60arc

In the figure: θ = 60° is one sixth of the whole circle, so the arc is 1/6 of the circumference.

Rule 1: Arc length — Take the fraction θ/360 of the full circumference.

  • Easy: r = 7, θ = 90°: (1/4) × 44 = 11.
  • Medium: r = 21, θ = 60°: (1/6) × 132 = 22.
  • Exam-type: Semicircle r = 14: 44.

Rule 2: Find the angle or radius — Rearrange: θ = l × 360 / (2πr).

  • Easy: Arc 11, r = 7: θ = 90°.
  • Medium: Sector perimeter: arc 11 + 2 × 7 = 25.
  • Exam-type: Arc 22, θ = 180°: r = 7.
Remember

A bigger angle at the centre means a longer arc, for the same radius.

Common mistake

Forgetting to add the two radii when asked for the perimeter of a sector.

Memory clue

"Slice fraction times the whole edge."

Practice questions

Easy
  1. 1A semicircular arc is what fraction of the full circumference?
  2. 2What fraction of the full circumference is an arc that subtends 72° at the centre?
  3. 3What fraction of the full circumference is an arc that subtends 45° at the centre?
Medium
  1. 4What fraction of the full circumference is an arc that subtends 90° at the centre?
  2. 5What fraction of the full circumference is an arc that subtends 270° at the centre?
  3. 6The length of an arc that subtends θ° at the centre of a circle of radius r is
Challenging
  1. 7A quarter-circle arc has radius 7 cm. Find its length. (Take π = 22/7.)
  2. 8A quarter-circle arc has radius 14 cm. Find its length. (Take π = 22/7.)
Show answers (try them yourself first)
  1. 1one half
  2. 20.2
  3. 30.125
  4. 40.25
  5. 50.75
  6. 6(θ/360) × 2πr
  7. 711 cm
  8. 822 cm

Area of rectangles and parallelograms

From daily life

Tiles needed to cover a floor depend on its area, the number of unit squares inside.

Simple definition

Area measures the surface inside a shape in square units. Rectangle l × b; square a²; parallelogram base × height (height is the perpendicular distance, not the slanted side).

Formula: Rectangle l × b; square a²; parallelogram b × h

base bhA = b × h

In the figure: The slanted side is not the height; the dashed perpendicular h is.

Rule 1: Find the area — Multiply the two perpendicular measures.

  • Easy: Rectangle 12 × 5 = 60 sq units.
  • Medium: Parallelogram b = 10, h = 6: 60.
  • Exam-type: Square side 7: 49.

Rule 2: Compare and convert — Same base and same height give equal areas. 1 m² = 10 000 cm².

  • Easy: Rectangle and parallelogram on base 8 with height 5: both 40.
  • Medium: Floor 4 m × 5 m with 50 cm square tiles: 20 ÷ 0.25 = 80 tiles.
  • Exam-type: Area 72, b = 9: h = 8.
Remember

Area is in square units (cm², m²); perimeter is in plain length units.

Common mistake

Using the slanted side instead of the perpendicular height.

Memory clue

"Base times straight-up height."

Practice questions

Easy
  1. 1A rectangle is 35 m by 21 m. Find its area.
  2. 2A parallelogram has base 7 cm and height 11 cm. Find its area.
  3. 3The area of a rectangle is 342 cm² and its length is 19 cm. Find its width.
Medium
  1. 4A rectangular lawn is 29 m by 11 m. A path 3 m wide surrounds the lawn on the outside. Find the area of the path.
  2. 5In a parallelogram, one side 14 cm has a corresponding height of 6 cm. The adjacent side is 12 cm. Find the height corresponding to that side.
  3. 6In a parallelogram, one side 9 cm has a corresponding height of 8 cm. The adjacent side is 12 cm. Find the height corresponding to that side.
Challenging
  1. 7A rectangle has a diagonal of 25 cm and one side 7 cm. Find its area.
  2. 8The length of a rectangle is increased by 10% and its width is decreased by 20%. What is the percentage change in the area?
Show answers (try them yourself first)
  1. 1735 m²
  2. 277 cm²
  3. 318 cm
  4. 4276 m²
  5. 57 cm
  6. 66 cm
  7. 7168 cm²
  8. 812% decrease

Area of a triangle and Heron's formula

From daily life

A triangle is half of a parallelogram made by joining two copies. When the height is hard to measure, the three sides are enough.

Simple definition

Area = ½ × base × height. With sides a, b, c and semi-perimeter s = (a + b + c)/2, Heron's formula: Area = √[s(s − a)(s − b)(s − c)]. Equilateral triangle: (√3/4)a².

Formula: A = ½bh; A = √[s(s−a)(s−b)(s−c)]

abcs = (a+b+c)/2A = √[s(s−a)(s−b)(s−c)]

In the figure: Use Heron's formula when you know a, b and c but not the height.

Rule 1: Base and height — Half of base times the perpendicular height.

  • Easy: b = 10, h = 6: 30.
  • Medium: Right triangle, legs 6 and 8: 24.
  • Exam-type: Area 40, b = 10: h = 8.

Rule 2: Heron's formula — Find s first, then the three differences.

  • Easy: 3, 4, 5: s = 6, area √(6·3·2·1) = 6.
  • Medium: 13, 14, 15: s = 21, area √(21·8·7·6) = 84.
  • Exam-type: Equilateral side 4: (√3/4)·16 = 4√3.
Remember

Sides must satisfy the triangle inequality: the two shorter sides add to more than the longest.

Common mistake

Using s = a + b + c instead of half of it.

Memory clue

"Half-perimeter first, then multiply the gaps."

Practice questions

Easy
  1. 1A triangle has base 6 cm and height 25 cm. Find its area.
  2. 2A right-angled triangle has perpendicular sides 7 cm and 11 cm. Find its area.
  3. 3Use Heron's formula to find the area of a triangle with sides 17 cm, 17 cm and 16 cm.
Medium
  1. 4The area of a triangle is 142.5 cm² and its base is 15 cm. Find the corresponding height.
  2. 5Find the area of an equilateral triangle of side 14 cm.
  3. 6A median of a triangle of area 110 cm² divides it into two triangles. What is the area of each?
Challenging
  1. 7A triangle has sides 5 cm, 5 cm and 6 cm (its area is 12 cm²). Find the altitude drawn to the side of length 5 cm.
  2. 8In quadrilateral ABCD, the diagonal AC = 25 cm. The perpendiculars from B and D to AC are 6 cm and 8 cm. Find the area of ABCD.
Show answers (try them yourself first)
  1. 175 cm²
  2. 238.5 cm²
  3. 3120 cm²
  4. 419 cm
  5. 549√3 cm²
  6. 655 cm²
  7. 74.8 cm
  8. 8175 cm²

Area of a circle and a sector

From daily life

A round pizza of radius 15 cm covers more than a square one of side 15 cm. A slice covers a fraction of it.

Simple definition

Area of a circle = πr². A sector of angle θ has area (θ/360) × πr². A ring between circles of radii R and r has area π(R² − r²).

Formula: A = πr²; sector = (θ/360)πr²; ring = π(R² − r²)

A = πr²

In the figure: The red slice is the fraction θ/360 of the full circle area.

Rule 1: Circle area — Square the radius first, then multiply by π.

  • Easy: r = 7: 154.
  • Medium: d = 20: r = 10, area 314 (π = 3.14).
  • Exam-type: Area 616: r² = 196, r = 14.

Rule 2: Sector and ring — Take the fraction for the sector; subtract inner from outer for a ring.

  • Easy: r = 14, θ = 90°: (1/4) × 616 = 154.
  • Medium: Semicircle r = 7: 77.
  • Exam-type: Ring R = 10, r = 7: π × 51 ≈ 160.1.
Remember

Doubling the radius makes the area four times as big.

Common mistake

Using πr × 2 (circumference) for area, or forgetting to square r.

Memory clue

"Area has a square: r squared."

Practice questions

Easy
  1. 1A sector has radius 28 cm and arc length 84 cm. Find its area using A = ½ × arc length × radius.
  2. 2A sector has radius 7 cm and arc length 14 cm. Find its area using A = ½ × arc length × radius.
  3. 3A sector has radius 14 cm and arc length 56 cm. Find its area using A = ½ × arc length × radius.
Medium
  1. 4A sector has radius 21 cm and arc length 105 cm. Find its area using A = ½ × arc length × radius.
  2. 5A sector has radius 7 cm and arc length 35 cm. Find its area using A = ½ × arc length × radius.
  3. 6The area of a circle of radius r is
Challenging
  1. 7The area of a sector with angle θ in a circle of radius r is
  2. 8Find the area of a circle of radius 21 cm. (Take π = 22/7.)
Show answers (try them yourself first)
  1. 11176 cm²
  2. 249 cm²
  3. 3392 cm²
  4. 41102.5 cm²
  5. 5122.5 cm²
  6. 6πr²
  7. 7(θ/360) × πr²
  8. 81386 cm²
Live figure

Area of a circle and a sector

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