Circle Area Formula Derivation Guide

MathematicsBeginnerReading time: 3 min

Overview

Have you ever wondered how people calculated the area of a circle before computers? This experiment takes you through the wisdom of Archimedes and Liu Hui, using the limit concept of 'exhaustion' and rearrangement to visually derive the circle area formula. We will cut a circle into countless small sectors and rearrange them into a familiar geometric shape, discovering mathematical truth from the unknown.

Background

As early as the 3rd century BC, the ancient Greek mathematician Archimedes used the 'Method of Exhaustion' to estimate Pi and the area of a circle. In China, mathematician Liu Hui of the Wei and Jin periods created the 'Circle Cutting Technique', stating 'The finer the cut, the smaller the loss. Cut again and again until it cannot be cut, then it becomes one with the circle and no loss remains.' Both methods use limits to transform curve problems into straight-line problems.

Key Concepts

Radius (r)

r

A line segment from the center of the circle to any point on its circumference.

Circumference (C)

C = 2\pi r

The distance around the circle. We know it is $\pi$ times the diameter.

Sector Cutting

\lim_{n \to \infty}

Dividing the circle into several congruent small sectors. As the number of divisions $n$ increases, the arc edge of the sector becomes closer to a straight line.

Rearrangement

\text{Area}_{circle} = \text{Area}_{rectangle}

An ancient geometric idea of transforming a circle into a rectangle or parallelogram of equal area through cutting and rearranging.

Experiment Steps

1
Observe Initial State
In the control panel, set the number of sectors $n$ to the minimum value of $4$ . Observe how many parts the circle is divided into. Imagine if these sectors were arranged in an alternating pattern, what shape would they form?
2
Initial Rearrangement
Click 'Start' or drag the 'Rearrange' slider. Observe how these sectors move and interlock. What does the resulting shape look like? Is the edge straight?
3
Infinite Approximation
Gradually increase the number of sectors $n$ , observing the effects at $n=16, 32, 64$ . As $n$ increases, what happens to the top and bottom edges of the shape? Which standard geometric shape does it increasingly resemble?
4
Derive Formula
When $n$ is large enough, we can view this shape as a Rectangle. Observe the labels: 1. What dimension of the circle does the height of the rectangle correspond to? 2. How much of the circumference is the width of the rectangle? Combining with the rectangle area formula $S = \text{Width} \times \text{Height}$ , can you write the formula for the area of a circle?

Learning Outcomes

Understand the limit concept in deriving the circle area.
Master the derivation process of the circle area formula $S = \pi r^2$ .
Recognize that as the number of cuts increases, the edges become straighter and the error becomes smaller.
Experience the process of mathematical modeling transforming geometric shapes.

Real-world Applications

Pizza Pricing: Why is a 12-inch pizza larger than two 6-inch pizzas combined? (Area is proportional to the square of the radius)
Land Surveying: Calculating the base area of circular granaries to estimate grain storage in ancient agriculture.
Architecture: Calculating material usage for modern circular buildings (e.g., stadiums, domes).
Medical Imaging: CT scans use integral principles (similar to this limit concept) to reconstruct circular cross-sectional images of the human body.

Common Misconceptions

Misconception

Misconception: The rearranged shape always has wavy edges and cannot be a rectangle.

Correct

Correction: Under the concept of mathematical limits, as the number of cuts approaches infinity, the difference between the arc and its chord approaches zero, so in the limit state, it is strictly a rectangle.

Misconception

Misconception: The width of the rectangle is the circumference.

Correct

Correction: Observe the arrangement of sectors; the top and bottom sides each take up half of the sectors. Therefore, the width of the rectangle is only half of the circumference (

\pi r

), not the whole circumference.

Ready to start?

Now that you understand the basics, start the interactive experiment!

Start Experiment Start Quiz