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Themed maths · 14 March

Pi Day maths activities for IGCSE and GCSE

14 March is Pi Day (3/14). This ready-to-teach lesson estimates π by experiment and compares famous fractions for π: a 5-minute starter, a 35-minute main activity, an extension and full worked answers.

Level
IGCSE and GCSE (Higher and Extended)
Time
40 minutes, plus a 10-minute extension
Topics
Circumference and area of circles; Relative frequency and expected outcomes; Percentage error; Fractions and decimals
Equipment
The starter is non-calculator. A calculator is allowed in the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minπ from a circle and a square
Main: task B12 minBuffon’s needle
Main: task C11 minFractions close to π
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator unless a question says so.

  1. Find the circumference of a circle of diameter 10 cm, in terms of π.
  2. Find the area of a circle of radius 4 cm, in terms of π.
  3. Write 22/7 as a decimal to 3 decimal places.
  4. Write π correct to 3 significant figures.

Main activity (35 minutes)

Task A: π from a circle and a square (12 min)

A circle of radius 10 cm is drawn inside a square of side 20 cm. Grains of rice are dropped at random onto the square.

  1. Find the area of the square and of the circle (to 3 significant figures).
  2. What fraction of the square is inside the circle? Give it in terms of π and to 3 significant figures.
  3. 500 grains land on the square and 391 are inside the circle. Find the relative frequency, and use it to estimate π.

Task B: Buffon’s needle (12 min)

Cocktail sticks 5 cm long are dropped onto paper ruled with lines 5 cm apart. The probability that a stick crosses a line is 2/π.

  1. Find the probability to 3 significant figures.
  2. 200 sticks are dropped. How many would you expect to cross a line?
  3. A class drops 200 sticks and 130 cross a line. Use this to estimate π, to 3 significant figures.

Task C: Fractions close to π (11 min)

Two famous fractions are close to π: 22/7 and 355/113.

  1. Write both fractions as decimals to 6 decimal places. Which is closer to π = 3.141593?
  2. Find the percentage error when 22/7 is used for π, to 3 significant figures.
  3. A bicycle wheel has diameter 70 cm. Using π = 22/7, how far does the bicycle travel in 100 turns of the wheel? Give your answer in metres.

Extension (10 minutes)

For fast finishers, or as homework.

  1. π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − … Use the first 5 terms to estimate π, to 3 significant figures.
  2. In the rice experiment, about how many grains would need to land on the square for 1000 to be expected inside the circle?

For teachers

Teacher notes and full worked answers
  • Pi Day is 14 March because 3/14, written month first, matches π = 3.14…
  • Pool the whole class’s results in tasks A and B: more trials give a better estimate.
  • In task C (b), students often divide by 22/7 instead of by π: ask what the error is ‘a percentage of’.

Starter

  1. 10π cm
    • π × 10
  2. 16π cm2
    • π × 42
  3. 3.143
    • 22 ÷ 7 = 3.142857…
  4. 3.14
    • π = 3.14159…

Task A: π from a circle and a square

  1. 400 cm2 and 314 cm2
    • 20 × 20 = 400
    • π × 102 = 100π = 314.15…
  2. π/4 ≈ 0.785
    • 100π ÷ 400 = π/4 = 0.7853…
  3. 0.782; π ≈ 3.128
    • 391 ÷ 500 = 0.782
    • 0.782 ≈ π/4, so π ≈ 4 × 0.782 = 3.128

Task B: Buffon’s needle

  1. 0.637
    • 2 ÷ π = 0.6366…
  2. 127
    • 200 × 2/π = 127.3…, so about 127.
  3. 3.08
    • 130/200 ≈ 2/π, so π ≈ 2 × 200 ÷ 130 = 3.076…

Task C: Fractions close to π

  1. 3.142857 and 3.141593: 355/113 is closer
    • 22 ÷ 7 = 3.142857…
    • 355 ÷ 113 = 3.1415929…
  2. 0.0402%
    • (22/7 − π) ÷ π × 100 = 0.04024…
  3. 220 m
    • One turn: 22/7 × 70 = 220 cm.
    • 100 turns: 22 000 cm = 220 m

Extension

  1. 3.34
    • 1 − 1/3 + 1/5 − 1/7 + 1/9 = 0.8349…
    • 4 × 0.8349… = 3.339…
  2. 1273
    • n × π/4 = 1000, so n = 4000/π = 1273.2…

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

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