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Themed maths · 14 March

Pi Day maths for CBSE Class 10

14 March is Pi Day (3/14). This ready-to-teach lesson looks at rational numbers close to π, estimates π by experiment and uses π = 22/7 in circle problems: a 5-minute starter, a 35-minute main activity, an extension and full worked answers. No calculator.

Level
CBSE Class 10 (Class 9 can try most of it)
Time
40 minutes, plus a 10-minute extension
Topics
Real numbers: rational and irrational numbers; Areas related to circles; Probability; Decimal expansions
Equipment
No calculator. Use π = 22/7 where a question says so.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A11 minFractions and π
Main: task B12 minπ from a dartboard
Main: task C12 minWheels and tracks
Extension10 minFast finishers or homework

Starter (5 minutes)

Quick-fire mental maths. Use π = 22/7.

  1. Find the circumference of a circle of radius 7 cm. (Use π = 22/7.)
  2. Find the area of a circle of radius 14 cm. (Use π = 22/7.)
  3. 22/7 = 3.142857142857… How many digits are in the repeating block?
  4. A point is chosen at random inside a square of side 2. What is the probability that it lies inside the circle of radius 1 drawn in the square? Give your answer in terms of π.

Main activity (35 minutes)

Task A: Fractions and π (11 min)

π is irrational: its decimal expansion never ends and never repeats. 22/7 and 355/113 are rational numbers close to it.

  1. Divide 22 by 7 by long division and write down the repeating block of digits.
  2. Find 355/113 − 22/7 as a single fraction.
  3. Find a rational number between 355/113 and 22/7 by adding the numerators and adding the denominators.

Task B: π from a dartboard (12 min)

A square board of side 28 cm has a circle of radius 14 cm drawn inside it, touching all four sides. Darts land at random on the board. Use π = 22/7.

  1. Find the area of the circle, the area of the square, and the probability that a dart lands inside the circle.
  2. 700 darts are thrown. How many would you expect inside the circle?
  3. In a class experiment, 546 of 700 darts land inside the circle. Use the exact probability π/4 to estimate π from this result.

Task C: Wheels and tracks (12 min)

Use π = 22/7.

  1. A wheel has radius 35 cm. How far does it roll in one turn? How many turns does it make in 1.1 km?
  2. A circular running track has inner radius 70 m and outer radius 77 m. Find the area of the track.
  3. How much further is one lap around the outer edge than around the inner edge?

Extension (10 minutes)

For fast finishers, or as homework.

  1. π/4 = 1 − 1/3 + 1/5 − 1/7 + … Use the first 4 terms to find an estimate of π as a fraction.
  2. A circle is drawn inside a square of side 14 cm, touching all four sides. Find the area of the square outside the circle. (Use π = 22/7.)

For teachers

Teacher notes and full worked answers
  • Pi Day is 14 March because 3/14, written month first, matches π = 3.14…
  • Task A links to the Real numbers chapter: 22/7 is rational (its decimal repeats), π is irrational.
  • Pool the class’s dart results in task B: more throws give a better estimate.

Starter

  1. 44 cm
    • 2 × 22/7 × 7 = 44
  2. 616 cm2
    • 22/7 × 14 × 14 = 616
  3. 6 (the block 142857)
    • The digits 142857 repeat for ever.
  4. π/4
    • Circle area π ÷ square area 4

Task A: Fractions and π

  1. 142857
    • 22 ÷ 7 = 3 remainder 1; then 10 ÷ 7, 30 ÷ 7, 20 ÷ 7, 60 ÷ 7, 40 ÷ 7, 50 ÷ 7 give 1, 4, 2, 8, 5, 7, and the remainder 1 comes back.
    • So 22/7 = 3.142857142857…
  2. −1/791
    • (355 × 7 − 22 × 113)/(113 × 7) = (2485 − 2486)/791 = −1/791
  3. 377/120
    • (355 + 22)/(113 + 7) = 377/120
    • Check: 355/113 = 3.14159…, 377/120 = 3.14166…, 22/7 = 3.14285…

Task B: π from a dartboard

  1. 616 cm2, 784 cm2, 11/14
    • 22/7 × 14 × 14 = 616; 28 × 28 = 784
    • 616/784 = 11/14
  2. 550
    • 700 × 11/14 = 550
  3. 3.12
    • 546/700 is about π/4, so π is about 4 × 546/700 = 2184/700 = 3.12

Task C: Wheels and tracks

  1. 220 cm; 500 turns
    • 2 × 22/7 × 35 = 220 cm
    • 1.1 km = 110 000 cm, and 110 000 ÷ 220 = 500
  2. 3234 m2
    • 22/7 × (772 − 702) = 22/7 × (77 − 70)(77 + 70) = 22/7 × 7 × 147 = 3234
  3. 44 m
    • 2π × 77 − 2π × 70 = 2π × 7 = 2 × 22/7 × 7 = 44

Extension

  1. 304/105
    • 1 − 1/3 + 1/5 − 1/7 = 76/105
    • 4 × 76/105 = 304/105 (about 2.9: this series needs many terms).
  2. 42 cm2
    • 196 − 22/7 × 7 × 7 = 196 − 154 = 42

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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