# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/circle-arc-sector/pi-history
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/circle-arc-sector/pi-history/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "History of Pi",
  description: "Discover pi's fascinating journey from ancient Babylonians to modern mathematics. Learn Archimedes' methods, Zu Chongzhi's breakthroughs, and cultural impact.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Circle Arcs and Sectors",
};

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

## Understanding and Meaning of Pi

Have you ever wondered why the circumference of a circle always has a constant relationship with its diameter? This amazing mathematical constant is called <InlineMath math="\pi" /> (pi). The value of <InlineMath math="\pi" /> represents the ratio between the circumference of a circle and its diameter, which always produces the same number for all circles, approximately 3.14159.

Mathematically, <InlineMath math="\pi" /> can be defined as:

<BlockMath math="\pi = \frac{\text{circle circumference}}{\text{circle diameter}}" />

This constant is very special because its value never changes, regardless of how big or small the circle is. Imagine it like a cake recipe that always produces the same taste even when the portion is enlarged or reduced.

## Early Discoveries of Ancient Civilizations

People in ancient times already recognized the special relationship between the circumference and diameter of a circle. The ancient Babylonian civilization used a simple approach by considering <InlineMath math="\pi" /> to have a value of 3. Although not accurate, this approach was sufficient for their practical needs in building structures and calculating land area.

The ancient Egyptians had a more sophisticated approach. They used the value <InlineMath math="\pi \approx \frac{256}{81}" />, which when calculated gives a result of about 3.16. This value was more accurate compared to the Babylonian approach and showed a deeper understanding of mathematics.

<div className="flex flex-col gap-4">
<BlockMath math="\frac{256}{81} = 3.160..." />
</div>

## Archimedes Method and Polygons

Archimedes of Syracuse, a Greek mathematician who lived around 287-212 BC, developed a revolutionary method to calculate <InlineMath math="\pi" /> with greater precision. He used an approach of regular polygons that surrounded and were inside the circle.

The basic concept was simple yet brilliant. Archimedes drew regular polygons inside and outside the circle, then calculated the perimeter of both polygons. The perimeter of the inner polygon provided a lower bound for the circle's circumference, while the perimeter of the outer polygon provided an upper bound.

<LineEquation
  title={<>Visualization of Archimedes Polygon Method</>}
  description="Regular polygons that surround and are inside the circle to approximate the value of pi."
  cameraPosition={[0, 0, 10]}
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 37 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 36;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0
        };
      }),
      color: getColor("CYAN"),
      smooth: true,
      showPoints: false,
      labels: [{ text: "Circle", at: 9, offset: [0.5, 0.5, 0] }]
    },
    {
      points: Array.from({ length: 7 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 6;
        return {
          x: 3 * Math.cos(angle),
          y: 3 * Math.sin(angle),
          z: 0
        };
      }),
      color: getColor("ORANGE"),
      smooth: false,
      showPoints: true,
      labels: [{ text: "Inner Polygon", at: 2, offset: [-1, -0.5, 0] }]
    },
    {
      points: Array.from({ length: 7 }, (_, i) => {
        const angle = (i * 2 * Math.PI) / 6;
        const radius = 3;
        const apothem = radius / Math.cos(Math.PI / 6);
        return {
          x: apothem * Math.cos(angle),
          y: apothem * Math.sin(angle),
          z: 0
        };
      }),
      color: getColor("PURPLE"),
      smooth: false,
      showPoints: true,
      labels: [{ text: "Outer Polygon", at: 4, offset: [0.8, -0.5, 0] }]
    }
  ]}
/>

Using a 96-sided polygon, Archimedes successfully determined that the value of <InlineMath math="\pi" /> lies between <InlineMath math="\frac{223}{71}" /> and <InlineMath math="\frac{22}{7}" />. His calculations gave:

<div className="flex flex-col gap-4">
<BlockMath math="3.1408 < \pi < 3.1429" />
</div>

## Contributions of Chinese Mathematicians

Zu Chongzhi, a Chinese mathematician who lived in the 5th century AD, achieved an extraordinary breakthrough in calculating <InlineMath math="\pi" />. He used a polygon with 24,576 sides and successfully determined the value of <InlineMath math="\pi" /> to seven accurate decimal places.

Zu Chongzhi found that <InlineMath math="\pi \approx \frac{355}{113}" />, which gives the value 3.1415929. This approximation was remarkable because it was accurate to six decimal places and was not surpassed for almost a thousand years.

<div className="flex flex-col gap-4">
<BlockMath math="\frac{355}{113} = 3.1415929..." />
</div>

## Modern Era and Pi Symbol

William Jones, a Welsh mathematician, first introduced the symbol <InlineMath math="\pi" /> in 1706 in his work "Synopsis Palmariorum Matheseos". The choice of this Greek letter was very appropriate because <InlineMath math="\pi" /> is the first letter of the word "perimeter" in Greek, which means circumference.

Leonhard Euler, the famous Swiss mathematician, popularized the use of the symbol <InlineMath math="\pi" /> through his influential works. Thanks to Euler, this symbol became the universal standard in mathematics to this day.

## Special Properties of Pi

The value of <InlineMath math="\pi" /> has very interesting characteristics. It is an irrational number, meaning it cannot be expressed as a simple fraction of two integers. Moreover, <InlineMath math="\pi" /> is also a transcendental number, which means it cannot be the root of a polynomial equation with rational coefficients.

In daily life, we often use approximations <InlineMath math="\pi \approx 3.14" /> or <InlineMath math="\pi \approx \frac{22}{7}" /> for practical calculations. However, for applications requiring high precision such as satellite technology or physics research, more decimal places are needed.

The fundamental relationship of <InlineMath math="\pi" /> with circle geometry can be expressed as:

<div className="flex flex-col gap-4">
<BlockMath math="C = \pi \cdot d" />
<BlockMath math="A = \pi \cdot r^2" />
</div>

where <InlineMath math="C" /> is circumference, <InlineMath math="d" /> is diameter, <InlineMath math="A" /> is area, and <InlineMath math="r" /> is the radius of the circle.