# What is the equation for a half a circle?

Apr 8, 2016

In polar co-ordinates, r = a and $\alpha < \theta < \alpha + \pi$.

#### Explanation:

The polar equation of a full circle, referred to its center as pole, is r = a. The range for $\theta$ for the full circle is $\pi$.

For half circle, the range for $\theta$ is restricted to $\pi$.

r = a and $\alpha < \theta < \alpha + \pi$, where a and $\alpha$ are constants for the chosen half circle.

Apr 8, 2016

In rectangular coordinates, the equation of the upper half of a circle can be written:

$y = \sqrt{{r}^{2} - {\left(x - h\right)}^{2}} + k$

#### Explanation:

The equation of a full circle with centre $\left(h , k\right)$ and radius $r$ can be written:

${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$

Hence the upper half of a circle can be expressed as:

$y = \sqrt{{r}^{2} - {\left(x - h\right)}^{2}} + k$

where $\left(h , k\right)$ is the centre and $r$ the radius.