Question

What is the equation for a half a circle?

Answer 1

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`.

So, the answer is
r = a and `alpha < theta < alpha + pi`, where a and `alpha` are constants for the chosen half circle.

Answer 2

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

`y = sqrt(r^2 - (x-h)^2) + k`

Explanation:

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

`(x-h)^2+(y-k)^2 = r^2`

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

`y = sqrt(r^2 - (x-h)^2) + k`

where `(h, k)` is the centre and `r` the radius.