Rearranging Formulae

The area of a circle is A = \pi r^2. That formula is built to answer one question: "I know the radius — what's the area?" But real life loves to hand you the problem the other way round. A gardener knows the circular flower bed covers 12 m² and needs the radius to buy edging. A formula written for A now has to give up r instead.

That is rearranging a formula — also called changing the subject. The letter standing alone is the subject; here it is A. To make r the subject you rewrite the formula so it is the one sitting alone. And the wonderful part: you already know every move. It is the exact same balance-scale logic you use when solving two-step equations — do the same thing to both sides — only now the "numbers" are letters.

The method: undo, in reverse, on both sides

Treat every other letter as if it were an ordinary number, and peel away whatever is done to your target letter — one inverse operation at a time, undoing the last thing first. Multiplied? Divide. Added? Subtract. Squared? Take the square root.

To make r the subject of the circumference formula C = 2\pi r: the r has been multiplied by 2\pi, so divide both sides by 2\pi:

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

It is substitution in reverse: instead of feeding numbers in, you keep the letters and isolate the one you want.

See it rearranged

Watch v = u + at become a = \dfrac{v - u}{t}, one inverse move per step: first undo the +\,u, then undo the \times\,t. Step through it.

Worked example 1 — make a the subject of v = u + at

This is a real physics formula: final speed v equals starting speed u plus acceleration a times time t. Suppose you want the acceleration.

Start: v = u + at. The last thing added to at is u, so undo it first — subtract u from both sides: v - u = at.
Now a is multiplied by t, so divide both sides by t:

a = \frac{v - u}{t}

Notice the answer is a formula, not a number — exactly what we wanted.

Worked example 2 — make r the subject of A = \pi r^2

Back to the gardener's flower bed. Here r is squared, so undoing needs one extra move.

Start: A = \pi r^2. The r^2 is multiplied by \pi, so divide both sides by \pi: \dfrac{A}{\pi} = r^2.
Now r is squared. The inverse of squaring is the square root, so square-root both sides:

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

For the 12 m² bed: r = \sqrt{12/\pi} \approx 1.95 m. (A radius can't be negative, so we keep the positive root — more on that below.)

Worked example 3 — the subject is trapped inside a bracket

Sometimes the letter you want is buried. Make C the subject of the Fahrenheit formula F = \tfrac{9}{5}C + 32.

Undo the +\,32 first — subtract 32 from both sides: F - 32 = \tfrac{9}{5}C.
Now C is multiplied by the fraction \tfrac{9}{5}. To undo, multiply both sides by its flip, \tfrac{5}{9}:

C = \frac{5}{9}\left(F - 32\right)

Multiplying by a fraction is just "divide by 9, times 5" — the same balancing move as always.

The answer stays a formula. Rearranging follows the same balance rules as solving a numeric equation — do the same to both sides — but there are no numbers to crunch at the end. If you make r the subject of A = \pi r^2, the answer is r = \sqrt{A/\pi}, a formula in the other letters. Do not expect a single tidy number to fall out.
Undo a square with a square root. This is the step students forget. To go from r^2 to r you must take \sqrt{\phantom{x}} of both whole sides, not just the r. Strictly a square root gives a \pm, but for a physical quantity like a radius, a length, or a time, you keep the positive root — you can't have -1.95 metres of flower bed.
Square-root the sum, not each piece. \sqrt{A/\pi} is fine, but \sqrt{a^2 + b^2} is not a + b. The root wraps the whole side.

Because one formula quietly contains many. Take the speed relationship d = st (distance = speed × time). Rearrange it and you get s = \dfrac{d}{t} to find a speed from a journey, or t = \dfrac{d}{s} to work out how long a trip will take. Same formula, three tools — whichever quantity is the unknown, you re-solve for it.

This is algebra's "one formula, many uses" superpower, and it runs the modern world. A physicist re-solves E = mc^2 for mass; a spreadsheet user re-solves a mortgage formula for the monthly payment; a programmer re-solves a physics equation for whichever variable the game needs this frame. Learn to rearrange, and every formula you meet becomes a whole family of them.

See it explained

Sal Khan solves a formula for one of its letters by doing the same to both sides.