First, you need to remember the regular circle equation: \(x^2 + y^2 = r^2\). That’s the one that makes a full, smug, perfect circle. But a semi circle? It’s the circle’s half-baked cousin. You get it by solving for \(y\), and then making a choice.
Solving for \(y\) gives you \(y = \pm \sqrt{r^2 - x^2}\). That plus-or-minus sign is the key. The plus gives you the top half (like a smiling mouth). The minus gives you the bottom half (like a frown or an arched bridge). Your choice.
So the equation for a semi circle is literally just \(y = \sqrt{r^2 - x^2}\) for the top half. Or \(y = -\sqrt{r^2 - x^2}\) for the bottom. That’s it. No magic. Just a square root with an attitude.
But wait–what about the dome?
You might be thinking, “What if I want a semi circle that’s rotated or sitting sideways?” Oh, you fancy pants. That’s a different equation, but same logic. For a left-facing or right-facing semi circle, you just swap \(x\) and \(y\). So it becomes \(x = \pm \sqrt{r^2 - y^2}\). See? Math is just trading places with your variables.
But let’s be real–most people care about the top half. Why? Because it looks like a rainbow. Or a half-eaten cookie. Or the arc of a perfectly thrown football. It’s everywhere. And now you can math it.
Semi Circle Equation Formula Centroid & Center Of Mass Of A Semicircle
