# What is the formula for the radius of a semi-circle?

Dec 30, 2015

If given the area:

The normal area of a circle is $A = \pi {r}^{2}$. Since a semicircle is just half of a circle, the area of a semicircle is shown through the formula $A = \frac{\pi {r}^{2}}{2}$. We can solve for $r$ to show an expression for the radius of a semicircle when given the area:

$A = \frac{\pi {r}^{2}}{2}$

$2 A = \pi {r}^{2}$

$\frac{2 A}{\pi} = {r}^{2}$

$r = \sqrt{\frac{2 A}{\pi}}$

If given the diameter:

The diameter, like in a normal circle, is just twice the radius.

$2 r = d$

$r = \frac{d}{2}$

If given the perimeter:

The perimeter of a semicircle will be one half the circumference of its original circle, $\pi d$, plus its diameter $d$.

$P = \frac{\pi d}{2} + d$

$P = \frac{\pi \left(2 r\right)}{2} + 2 r$

$P = r \left(\pi + 2\right)$

$r = \frac{P}{\pi + 2}$

Note: by no means should you commit to memorizing the area or perimeter formulas I've derived here. While they could help you get to answer 30 seconds quicker, they are easily found if you just use logic! This is more an exercise of critical thinking and algebraic manipulation while expanding on your original knowledge of circles.