Chapter 24

Cartesian and polar

co-ordinates

24.1 Introduction

There are two ways in which the position of a point in
a plane can be represented. These are

(a) Cartesian co-ordinates, i.e. (x,y).

(b) Polar co-ordinates, i.e. (r,θ), whereris a radius
from a fixed point andθis an angle from a fixed
point.

24.2 Changing from Cartesian to

polar co-ordinates

In Figure 24.1, if lengthsx and y are known then
the length ofrcan be obtained from Pythagoras’ the-
orem (see Chapter 21) sinceOPQ is a right-angled
triangle.

y

P

O x Q x

r y



Figure 24.1

Hence,r^2 =(x^2 +y^2 ), from which r=

√

x^2 +y^2

From trigonometric ratios (see Chapter 21), tanθ=

y

x

from which θ=tan−^1

y

x

r=

√

x^2 +y^2 andθ=tan−^1

y

x

are the two formulae

we need to change from Cartesian to polar co-ordinates.

The angleθ, which may be expressed in degrees or radi-

ans, mustalwaysbe measured from the positivex-axis;

i.e., measured from the lineOQin Figure 24.1. It is

suggested that when changing from Cartesian to polar

co-ordinates a diagram should always be sketched.

Problem 1. Change the Cartesian co-ordinates

(3, 4) into polar co-ordinates

A diagram representing the point (3, 4) is shown in

Figure 24.2.

P

4

3

y

O x

r



Figure 24.2

From Pythagoras’ theorem, r=

√

32 + 42 =5 (note

that−5 has no meaning in this context).

By trigonometric ratios, θ=tan−^1

4

3

= 53. 13 ◦ or

0.927rad.

Note that 53. 13 ◦= 53. 13 ×

π

180

rad = 0.927rad.

DOI: 10.1016/B978-1-85617-697-2.00024-7