The area of a circle with radius r is given by the formula A = πr². This formula can be proven using various methods, but a common and intuitive approach involves rearranging sectors of the circle into a shape resembling a rectangle.
Proof by Rearranging Sectors
This method demonstrates the area of a circle by dividing it into many small sectors (like slices of a pie) and then rearranging these sectors to form a shape that approximates a rectangle.
Steps of the Proof
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Divide the Circle: Imagine a circle is divided into a large number of equal sectors.
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Rearrange the Sectors: Arrange these sectors side-by-side, alternating the point of each sector up and down.
- Half of the sectors are placed with their points facing upwards.
- The other half are placed with their points facing downwards, fitting into the gaps left by the first set of sectors.
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Forming a Rectangle: As the number of sectors increases and becomes infinitely large, the shape formed by arranging these sectors gets closer and closer to a perfect rectangle.
- The curved edges of the sectors become nearly straight lines.
- The pointed ends of the sectors form the short sides (height) of the rectangle.
- The bases of the sectors (parts that were originally on the circle's circumference) form the long sides (base) of the rectangle.
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Determining the Dimensions:
- Height: The "height" of this approximate rectangle is equal to the radius (r) of the original circle, as this is the length from the center to the edge of each sector.
- Base: The "base" of this approximate rectangle is formed by half of the circle's circumference. The total circumference of a circle is 2πr. Since half of the sectors form one long side and the other half form the opposite long side, each long side has a length equal to half the circumference.
- Half the circumference = (1/2) * 2πr = πr.
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Calculating the Area: The area of a rectangle is calculated by multiplying its base by its height.
- Using the dimensions derived from the rearranged circle sectors:
- Base = πr
- Height = r
- Area of the rectangle = Base * Height
- Area = πr * r
- Using the dimensions derived from the rearranged circle sectors:
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Combining Terms: Combining the r terms:
- Area = πr²
This matches the information from the reference which states, "Out and base is equal to Pi R so now 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 confirms the base is πr and the height is r, resulting in an area of πr².
Since the area of the rectangle formed by rearranging the sectors of the circle is πr², and this rectangle was formed from the entire circle, the area of the circle must also be πr².
This method provides a strong visual and intuitive proof for the formula of the area of a circle.
