Changing the subject of a formula

Sometimes you need to rearrange the formula to find the value you're looking for.

Example - circles

The area of a circle ( {A}) is \pi {r}^{2}.

A = \pi {r}^{2}

This is useful if we know the radius of the circle and want to know the area. But what if we want to find the radius?

We need a formula that has {r}= (some expression in {A}). We get this by rearranging the A = \pi r^2 formula like this:

Start by dividing both sides by \pi:


Then find the positive square root of both sides:


Swap the sides to make it easier to read:


The formula has been rearranged. We say that now r is the subject.


Q1. The formula to find the circumference of a circle: C = 2 \pi r

Rearrange the formula to make {r} the subject.

Q2. The formula to convert the temperature in ^\circ{F} to the temperature in ^\circ{C} is:

c = \frac {5(f - 32)} {9}

Rearrange this formula so that it will convert the temperature in ^\circ{C} to the temperature in ^\circ{F} (so that {f} is the subject).

A1. C = 2 \pi r, so divide both sides by 2 \pi:

\frac {C}{2 \pi} = r


r = \frac {C}{2 \pi}

A2. c = \frac {5(f - 32)} {9}

multiply by {9}:

9c = 5(f - 32)

divide by {5}:

\frac {9c} {5} = f - 32

add {32}:

\frac {9c} {5} + 32 = f


f = \frac {9c} {5} + 32

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