Physicist: To show this you generally have to usage either calculus or oranges. They both use much more or less the same ideas, they’re just applied in different ways.

You are watching: Why is the area of a circle pi r squared

Oranges:


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The pertinent fruit framework what makes the math happen.


Imagine taking an orange wedge and opening it so the the triangles all suggest “up” rather of towards the same point. If girlfriend interlaced 2 of these climate you’d have a tiny brick that’s about rectangular.


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By slicing the circle through thinner and also thinner triangles, the lines on the “wavey shape” all straighten out right into a rectangle.


As much more triangles are used, the curved finish produces less pronounced bumpiness and the directly sides come closer and closer come being right up and also down, make the brick rectangular. The elevation becomes same to the radius, when the size is half of the one (C = 2πR) which currently finds chin running along the top and also bottom. Together the variety of triangles “approaches infinity” the circle can be taken apart and also rearranged to fit almost perfectly into an “R through πR” box through an area of πR2.

This is why calculus is so damn useful. We regularly think the infinity as being secret or daunting to work-related with, however here the unlimited slicing simply makes the conclusion infinitely clean and exact: A = πR2.

Calculus:

On the mathier side of things, the one is the differential that the area. That is; if you boost the radius through “dr”, which is a tiny, small bit, then the area increases by Cdr where C is the circumference. We deserve to use that truth to define a disk together the amount of a many of an extremely tiny rings. “The sum of a many tiny _____” makes mathematicians reflexively say “use an integral“.


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With calculus, friend can find the area that a circle by adding up the areas of a lot of (infinite number) of an extremely thin (infinitely thin) rings. Below a random intermediate ring is red.


Every ring has actually an area that Cdr = (2πr)dr. Including them up from the center, r=0, to the outer edge, r=R, is written:

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This is a beautiful example of expertise trumping memory. A mathematician will forget the equation because that the area that a circle (A=πR2), yet remember that the circumference is its differential. That’s no to excuse their forgetfulness, just define it.

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