Proof relating to Measurement of angle The question asks to prove that radian is a constant angle. First, I didn't understand the question and what is way to prove that the radian is a constant angle.

Let $O$ be the centre of a circle and radius $OR = r$. If we take an arc $AB = OA = r$, then by definition, $∠AOB =1$ radian. Let $AO$ be produced to meet the circle at the point $C$. Then the length of the arc $ABC$ half the circumference and $∠AOC$, the angle at the centre subtended by this arc $= a$ straight angle $=$ two right angles.

Now if we take the ratio of the two arcs and that of the two angles, we have

arc $AB/$arc $ABC = r/(1/2 × 2\cdot\pi\cdot r) = 1/ \pi$

$∠AOB/∠AOC = 1$ radian$/2$ right angles

But in geometry, we can show that an arc of a circle is proportional to the angle it subtends at the centre of the circle.

Therefore, $∠AOB/∠AOC =$ arc $AB/$arc $ABC$

or, $1$ radian$/2$ right angles $= 1/\pi$

Therefore, $1$ radian $= 2/\pi$ right angles

This is constant as both $2$ right angles and $\pi$ are constants.

The approximate value of $\pi$ is taken as $22/7$ for calculation

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