what is area of one radian angle of a circle

[O. W. Grant]

Hi,

given:
a circle, radius = r;
angle of one radian.

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Radian
https://en.wikipedia.org/wiki/Radian

A radian on a circle has sides: a radius, a radius, an arc with length 1 radius.
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(1) Area of the circle = π.r^2

A circle has 2.π. rad.

If
area of 1 radian angle =1 rad
then
(2) area of the circle = 2.π.rad


(1), (2) => 2.π.rad = π.r^2 <=> 2. rad = r^2

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

 

[James R]

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
$$A=A'\times \frac{\theta}{2\pi} = \pi r^2 \times \frac{\theta}{2\pi} =\frac{1}{2}r^2\theta$$.

 

[James R]

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

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.