Exact Calculations with Pi

Why leave π in an answer?

For every circle, the distance around it divided by the distance across its centre is the same number: π. Its decimal begins $3.14159\ldots$ and never ends or repeats. Writing $3.14$ replaces π with a rounded value, whereas $8\pi$ gives an exact value. A multiple of π has a number multiplying it: in $8\pi$, that number is $8$.

Calculate with multiples of π

Treat π as a common factor when adding or subtracting its multiples. For example, $3\pi+5\pi=(3+5)\pi=8\pi$. Similarly, $7\pi-9\pi=-2\pi$. The same idea works with fractional : . The fraction work is ordinary fraction arithmetic; π stays in the answer.

You can also multiply or divide a multiple of π by a number. For instance, $4\times 3\pi=12\pi$ and $\frac{9\pi}{3}=3\pi$. Keep track of whether π itself is being multiplied: , not .

An ordinary number and a multiple of π are not like terms. You can write $3+2\pi$ exactly, but you cannot combine it into $5\pi$.

An exact answer for a circle

The circumference $C$ is the distance around a circle. It is $\pi$ times the diameter $d$, so $C=\pi d$. If a circle has diameter $7$ cm, its circumference is exactly $7\pi$ cm. Since the diameter is twice the radius , the same formula can be written . For a radius of cm, cm. Only convert to a decimal if the question asks for an approximation.

Exact form

π is not $3.14$: $3.14$ is an approximation. Leave answers as multiples of π when an exact answer is required.

Arithmetic

Combine coefficients of like π terms: $a\pi+b\pi=(a+b)\pi$. Apply ordinary fraction rules to fractional coefficients. Multiplying two π terms gives .