# 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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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

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