Distance Between Two Polar Points

In Cartesian coordinates the distance formula is short and familiar. In polar coordinates it is longer — a cosine appears — but the reasoning behind it is simple, and one property of cosine makes it easier to use than it looks.

Concept Two formulas, two systems

Cartesian, given

and

Polar, given

and

— longer, because of the cosine.

Neither system is better in general. For finding a distance the Cartesian form is simpler; for other tasks the polar form wins. Use whichever matches the coordinates you were given.

Theorem The polar distance formula

The two radii and the joining segment form a triangle. The angle between the radii is the difference of the two angles, and the formula is the cosine rule applied to that triangle.

Read it in three parts: a square root, the sum of the two squared radii, then a correction term that subtracts .

Example An exact answer

Find the distance between and .

Step 1 — the values:

Step 2 — the squares:

Step 3 — the product:

Step 4 — the angle difference:

, and

Step 5 — put it together:

Step 6 — take the root:

⟹ about 5.29 units

Example One that needs a calculator

Find the distance between and .

, and

⟹ about 4.87 units

Most angle differences do not give a neat cosine, so a calculator finishes the job.

Note The order of subtraction does not matter

Cosine is an even function, which means a negative angle gives the same value as the positive one:

So for the first example either subtraction works:

⟹ the same distance either way

This is worth knowing: it removes any worry about which angle to write first.

Summary

The polar distance formula is d = √(r₁² + r₂² − 2r₁r₂cos(θ₂ − θ₁)).
It is the cosine rule for the triangle made by the two radii.
Square the radii, add them, then subtract the cosine correction term.
Cosine is even, so cos(−θ) = cos(θ) and the order of subtraction is free.
For (4, 20°) and (6, 80°) the distance is √28 ≈ 5.29.