# How to express a circumference of half circle with this angle θ?

So this is my homework:

$$PQ=2$$ and $$PQ$$ is also the radius of the half-circle, $$QR$$ is parallel to $$PS$$, $$P$$ is the origin,

How do we express the circumference in terms of $$\theta$$? I still can't imagine how the size $$\theta$$ affects the circumference. The question asks to prove that $$K = 2 + 6 \cos(\theta) + 2 \sin (\theta)$$ where $$K$$ is the circumference of this half circle.

Thank you very much!

• What is $K$? I don't see it specified anywhere? Oct 17, 2020 at 2:35
• I don’t see a definition for $K$ in the image. Oct 17, 2020 at 2:35
• You will get a better response from mathSE reviewers if you edit your query to show your work. What have you tried? Where are you specifically having trouble? Oct 17, 2020 at 2:36
• To link $\theta$ with the circumference, I have to ask you some questions about your math background. What is the circumference of a circle of radius 2? Do you measure angles in degrees or radians? Do you understand the coordination between arc length (which is actually a fraction of 1 complete revolution = 1 complete circumference) and the angle which may be construed to be a fraction of 360 degrees or a fraction of $2\pi$ radians? It is difficult to help without knowing your background here. Oct 17, 2020 at 2:39
• Just checking: is $P$ not the origin, and should the question say $QR$ is parallel to $PS$? Make sure you check your question so that it makes sense to someone reading it for the first time. The more context you include, the better your question. Oct 17, 2020 at 3:01

The circumference, or the perimeter of $$PQRS$$ in terms of the given radius $$|PQ|=r=2$$ and the value of angle $$\theta$$ can be found as follows:
\begin{align} K&=|PQ|+|QR|+|RS|+|PS| \tag{1}\label{1} ,\\ |QR|&=2\,|QT| \tag{2}\label{2} ,\\ |QT|&=|PS|=r\cos\theta \tag{3}\label{3} ,\\ |RS|&=r\,\sin\theta \tag{4}\label{4} , \end{align}
so, indeed, we have \begin{align} K&=r+2r\cos\theta+r\,\sin\theta+r\cos\theta =r+3r\cos\theta+r\,\sin\theta \\ &=2+6\cos\theta+2\,\sin\theta \tag{5}\label{5} . \end{align}