Hi, given: a circle, radius = r; angle of one radian. - - - Radian https://en.wikipedia.org/wiki/Radian A radian on a circle has sides: a radius, a radius, an arc with length 1 radius. - - - (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?
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 \(1/\sqrt{2}=0.707\). ---- * 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 \(A=A'\times \frac{\theta}{2\pi} = \pi r^2 \times \frac{\theta}{2\pi} =\frac{1}{2}r^2\theta\).
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.
