The area of a circle can be proven to be πr² using a method that approximates the circle as a series of increasingly thin rectangles.
Understanding the Proof
The following explains the method used to prove the area of a circle:
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Divide the Circle: Imagine dividing the circle into many equal sectors.
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Rearrange the Sectors: Rearrange these sectors to form a shape that resembles a rectangle. The more sectors you divide the circle into, the closer the shape gets to being a perfect rectangle.
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Rectangle Dimensions: According to the YouTube video "Area of a circle, formula explained," as the sectors become infinitely small, the "rectangle" has the following properties:
- Height: The height of the rectangle is the radius (r) of the original circle.
- Base: The base of the rectangle is half of the circumference of the circle, which is πr. (The video mentions "base is equal to Pi R").
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Area Calculation: The area of a rectangle is base × height. Therefore, the area of this "rectangle" is πr × r = πr². The YouTube video states, "base * height becomes pi r * R combine the RS. Together and we have < R 2 which is equal to the area of the rectangle." This means:
Area ≈ base height = πr r = πr²
Formula
| Property | Description |
|---|---|
| Area of Rectangle | base * height |
| base | πr |
| height | r |
| Formula | πr * r = πr² |
In essence, by dissecting and rearranging the circle into a near-rectangular shape, we can use the area formula of a rectangle to derive and prove that the area of a circle is indeed πr².
