The Area Of A Circle Formula - Simple Intuitive Explanation

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We all know the area of a circle is pi times the radius squared. But where does that formula come from? Here is a visual animation that shows how a circle can be rearranged into a rectangle whose area is easy to calculate.

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There is another great explanation that MinutePhysics did where you can unroll a circle into a triangle, with the center becoming the tip of the triangle and the circumference becoming the base.

Since the area of a triangle is 1/2 * base * height, the base of the triangle is the circumference (2*pi*r) and the height is r, the area is 1/2 * 2 * pi * r * r = pi * r^2

Uejji

I’ve been looking all over google and this was the best explanation

melodyarmani

This is the simplest yet informative explanation i found. I randomly thought about why we use Radius^2 when finding volume instead of Diameter itself.

illuminaughty-

The way I think of it is the radius squared is a square. Multiplied by 4 is a square twice as big on its sides, but pi cuts a bit off the corners, to make it a circle

wiliqmh

the area of a circle is equal to the sum of circumferences of infinitely small rings .
let the radius be R
so, area = integration of 2πr (from 0 to R)
= 2π * integration of r (from 0 to R)
=2π*r^2/2 (from 0 to R)
=πR^2-π0^2
=πR^2

rabindranathghosh

I think that τ is a more intuitive constant than π, even in this case.
If you derive the area of any circle sector from the definition of the radian then you get a formula of the form A = 0.5⋅θ⋅r², which has the exactly same form as the full circle area in terms of τ, which is A = 0.5⋅τ⋅r².

Peter_

this has got to be the lamest comment section ever

hotdogskid

The best of the many intuitive way to explain the area of a circle.

FredoCorleone

I was watching another video on how was the formula created for area of circle then they never explain how the base is half of the circumference and this video explained it in just one sentence

Mikeshwartz

0:20 It is a parallelogram, not a rectangle
thus the height is not circumference/2

wada-wada

Wait, you didn't explain why the circumference is 2πr. without that, it's just a rearrangement.

arvindm

The curve lines are not equal to the straight one. All the dismental parts of a circle are built up as a pallarellogram but examine the both long sides of it which are made up by curve lines. I see they are not equal to the sides of the common parallelogram. So i think there is a mistake at the start of the process.

linhgabrielnguyen