If A = \(\frac{\theta}{360}\)\(\pi r^2\), make \(\theta\) the subject of the formula

The goal is to rearrange the formula so that \(\theta\) is on its own on one side. The given formula is:

\[ A = \frac{\theta}{360} \pi r^2 \]

Let's break down the steps to solve for \(\theta\):

• First, notice that \(\frac{\theta}{360}\) is being multiplied by \(\pi r^2\). To isolate \(\frac{\theta}{360}\), divide both sides by \(\pi r^2\):

\[ \frac{A}{\pi r^2} = \frac{\theta}{360} \]

• Now, to solve for \(\theta\), multiply both sides of the equation by 360:

\[ \theta = 360 \left(\frac{A}{\pi r^2}\right) \]

\[ \theta = \frac{360A}{\pi r^2} \]

This represents \(\theta\) in terms of A, r, and \(\pi\).

Why this method is correct:

• You use algebra to isolate the variable you want (here, \(\theta\)).
• You perform inverse operations to "undo" multiplication and get \(\theta\) alone.
• The answer must keep all variables and constants in the correct places, as determined by these algebraic steps.

Summary: The correct subject is

\[ \theta = \frac{360A}{\pi r^2} \]

This means that the angle \(\theta\) is equal to \(360\) multiplied by the area divided by \(\pi r^2\). This makes sense since you are calculating what fraction of the full circle (360 degrees) the sector's area represents.