Archimedes' Polygon
As the number of sides grows, the jagged perimeter closes in on the circle. This is the idea of a limit in plain sight.
Mathematics · Lesson 06
Pi & Circles
Pi is one of the most famous numbers in all of mathematics — and it shows up everywhere, from the orbit of planets to the waves in your earbuds. In this lesson you'll discover what Pi really is, why it never ends, and how to use it.
What is Pi (π)?
Pi is the ratio of a circle's circumference (the distance around) to its diameter (the distance across). No matter how big or small the circle, this ratio is always the same number:
π
≈ 3.14159 265 358…
circumference ÷ diameter = π, always
An irrational number — its decimal digits go on forever without repeating or ending.
Take any circle — a coin, a planet, a wheel. Divide its circumference by its diameter. You always get π. This is not a coincidence or a rule someone invented. It is a fundamental truth about the geometry of curved space.
Circumference vs. Diameter — a circle with diameter 1 has circumference π
The Two Essential Formulas
Once you know Pi, you can calculate anything about a circle using just the radius (r) — the distance from center to edge, which is half the diameter.
Circumference
C = 2πr
or C = πd (diameter × π)
Area
A = πr2
radius squared, then multiply by π
Solving step by step
- 1Identify what you know: radius, diameter, or circumference.
- 2If you have the diameter, halve it to get the radius (r = d ÷ 2).
- 3Plug into the formula. Use π ≈ 3.14159 or the π key on a calculator.
- 4Round your answer to the precision the problem asks for.
Archimedes Discovery Lab
Watch a circle unroll its own proof
Instead of memorizing 3.14, roll the circumference out onto a ruler, squeeze a circle with polygons, and rearrange area slices into a rectangle. The constant becomes visible.
Current Ratio (C/d)
3.14159
πd
d = 1.00
Diameter d1.00
Circumference C3.14159
One Full Turn0.00000
Area πr²0.78540
Slices to Rectangle
Cut a circle into more and more wedges, then alternate them. The shape approaches a rectangle with width πr and height r.
The Earth Test
If Earth's diameter is about 12,742 km, the distance around the equator follows the same rule as a coin or a wheel.
Using C = πd
40,030 km
Why precision matters
NASA only needs about 15 decimal places of π for mission work. Even that is enough to measure enormous circles with hair-thin accuracy.
A 4,000-Year History
Humans have been trying to pin down Pi for longer than written history. Each era got a little closer to the infinite truth.
- ~1900 BCE Babylonian mathematicians approximated Pi as 25/8 = 3.125, and Egyptian mathematicians in the Rhind Papyrus used (16/9)² ≈ 3.1605.
- ~250 BCE Archimedes of Syracuse squeezed Pi between two polygons — one inside the circle, one outside — calculating it lies between 3 10/71 and 3 1/7. His method was so clever it remained the gold standard for 1,500 years.
- ~480 CE Chinese mathematician Zu Chongzhi calculated Pi to 7 decimal places (3.1415926), a record that stood for nearly 900 years.
- 1706 Welsh mathematician William Jones first used the Greek letter π (pi) to represent the ratio. Euler popularised the symbol later, and it has been universal ever since.
- 1914 Indian genius Srinivasa Ramanujan, largely self-taught, derived extraordinarily fast-converging infinite series for Pi — series that are still used inside modern Pi-computing software.
- 2024 Using cloud computing, researchers calculated Pi to over 105 trillion digits — more digits than there are atoms in your body. Yet the digits never repeat and show no pattern.
The Digits of Pi
Pi never ends, so this lesson now includes a live search tool. Look for a birthday, zip code, or favorite pattern inside the first 100,000 digits and jump right to it.
Showing the opening digits of π.
Pi is irrational, and it is also believed to be normal. Over enough digits, every short pattern should appear somewhere — which is why digit search feels a little like hiding and seeking in infinity.
Pi in the Real World
Pi doesn't stay on paper — it is woven into physics, engineering, and technology everywhere you look.
Sound & Light Waves
Pi appears in the equations describing all wave motion — sound, light, radio, and ocean waves. Your earbuds and phone antenna both depend on it.
Space Navigation
NASA uses Pi to calculate orbital paths, planetary positions, and antenna dish sizes for deep-space communication with probes like Voyager.
Engineering & Design
Gears, pipes, tunnels, wheels, and domes — any circular or cylindrical structure requires Pi to calculate materials, forces, and dimensions.
DNA & Biology
The DNA double helix is a spiral, and Pi appears in equations describing its geometry. Pi also shows up in statistical models used in medical research.
Practice Problems
Use π ≈ 3.14159. Round answers to one decimal place unless it works out exactly.
1. A circle has radius r = 7. What is its circumference? (C = 2πr)
Hint: C = 2 × 3.14159 × 7 = 43.98…
2. A circle has radius r = 5. What is its area? (A = πr²)
Hint: A = 3.14159 × 5² = 3.14159 × 25 = 78.54…
3. A wheel has diameter d = 10 inches. How far does it travel in one full rotation? (C = πd)
Hint: C = 3.14159 × 10 = 31.4159…
4. A circular pizza has diameter 12 inches. What is its area? (Use r = d ÷ 2 first)
Hint: r = 6, A = 3.14159 × 36 = 113.1…
5. A circular running track has circumference 400 meters. What is its radius? (r = C ÷ 2π)
Hint: r = 400 ÷ (2 × 3.14159) = 400 ÷ 6.2832 ≈ 63.7