Distance Between Two Polar Points

The polar distance formula as the cosine rule on the triangle made by two radii - squaring the radii, subtracting the cosine correction term, and why the order of subtracting the angles never matters.

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Distance Between Two Polar Points — Moosa Academy

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 (x_1, y_1) and (x_2, y_2) :
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Polar, given (r_1, \theta_1) and (r_2, \theta_2) — 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
θ₂−θ₁ r₁ r₂ d pole
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.
d = \sqrt{\;r_1^{\,2} + r_2^{\,2} - 2 r_1 r_2 \cos(\theta_2 - \theta_1)\;}

Read it in three parts: a square root, the sum of the two squared radii, then a correction term that subtracts 2 r_1 r_2 \cos(\theta_2 - \theta_1) .

Example An exact answer

Find the distance between (4, 20^\circ) and (6, 80^\circ) .

Step 1 — the values: r_1 = 4 , \theta_1 = 20^\circ , r_2 = 6 , \theta_2 = 80^\circ
Step 2 — the squares: 4^2 + 6^2 = 16 + 36 = 52
Step 3 — the product: 2 \times 4 \times 6 = 48
Step 4 — the angle difference: 80^\circ - 20^\circ = 60^\circ , and \cos 60^\circ = 0.5
Step 5 — put it together: 52 - 48(0.5) = 52 - 24 = 28
Step 6 — take the root: \sqrt{28} \approx 5.29
⟹ about 5.29 units
Example One that needs a calculator

Find the distance between (3, 40^\circ) and (5, 110^\circ) .

3^2 + 5^2 = 9 + 25 = 34
2 \times 3 \times 5 = 30
110^\circ - 40^\circ = 70^\circ , and \cos 70^\circ \approx 0.342
34 - 30(0.342) \approx 34 - 10.26 = 23.74
\sqrt{23.74} \approx 4.87
⟹ 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:

\cos(-\theta) = \cos(\theta)
So for the first example either subtraction works:
\cos(80^\circ - 20^\circ) = \cos(60^\circ) = 0.5
\cos(20^\circ - 80^\circ) = \cos(-60^\circ) = 0.5
⟹ the same distance either way

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

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