# Prove that the length of any parallel of latitude is equal to the length of the equator times the cosine of the latitude angle.

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### 1 Answer

Assume as an approximation that the earth is a perfect sphere.

Call the radius of the sphere r.

Using the formula for the circumference of a circle, the length of the equator is `2pir`

Now, draw a triangle with a point at the centre of the earth A, a point on a particular latitude line B and a point horizontally across from that directly above the centre of the earth C. This is a right-angled triangle.

b

B_ _ _ C

\`theta` ) |

r \ |

\ |

_ _( `theta` \| A (centre of earth)

Call the length of the line BC 'b'. This is the radius of the circle round the earth at ``the latitude of B. The length of the line AB is the radius of the earth, r.

Call the latitude angle (the angle between the horizontal and the line AB) `theta` .

Using the z-angle rule of triangles, the angle between the lines AB and BC is also equal to `theta`

Now, `cos (theta) = b/r`

So the length of the equator `2pir` x `cos(theta)` is equal to

```2pir (b/r)= 2 pi b`

which is the length of the circumference of the earth at latitude `theta`