# what is area of one radian angle of a circle

Discussion in 'Physics & Math' started by O. W. Grant, Jun 24, 2022.

1. ### O. W. GrantRegistered Senior Member

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266
Hi,

given:

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(1) Area of the circle = π.r^2

If
then
(2) area of the circle = 2.π.rad

The area of two radians makes (the area of) one square?

3. ### James RJust this guy, you know?Staff Member

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38,678
The area of a 1 radian segment of a circle is $A=\frac{1}{2}r^2 \theta=\frac{1}{2}r^2(1)=\frac{1}{2}r^2$.*

Obviously, the area depends on the radius of the circle (actually the square of the radius).

Suppose the radius of the circle is r=1 unit. Then the area of a 1 radian segment will be 0.5 square units. This is equivalent to the area of a square with side length $1/\sqrt{2}=0.707$.

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* This formula can be derived as follows:
The area of a (full) circle is $A'=\pi r^2$.
A $\theta$ radian segment covers a fraction of $\theta/2\pi$ of that area because there are $2\pi$ radians of angle in a full circle.
Therefore, the area of a segment with angle $\theta$ is
$A=A'\times \frac{\theta}{2\pi} = \pi r^2 \times \frac{\theta}{2\pi} =\frac{1}{2}r^2\theta$.

Last edited: Jun 27, 2022
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5. ### James RJust this guy, you know?Staff Member

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38,678
Oh, I forgot. The area of a circle segment with radius r=1 unit and angle 2 radians is 1 square unit, which is the same as the area of a square with side length 1 unit, so what you said is correct about this particular example.

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