Area Comparison of Earth Strips by Latitude

The area of a strip of constant width around the Earth depends on its location relative to the equator. The Earth is approximately spherical.

Strip Area and Latitude Relationship

Consider a strip of a fixed width '$w$' running parallel to the equator at a certain latitude '$\phi$'. The area of such a strip is approximately equal to the circumference of the circle of latitude at that point multiplied by the strip's width.

The circumference of a circle of latitude '$\phi$' is given by $C_\phi = 2 \pi R \cos(\phi)$, where '$R$' is the Earth's radius.

Therefore, the area '$A_\phi$' of the strip is approximately:

$ A_\phi \approx C_\phi \times w = (2 \pi R \cos(\phi)) \times w $

Since the width '$w$' (10 m) and the Earth's radius '$R$' are constant, the area '$A_\phi$' is directly proportional to $\cos(\phi)$.

Comparing Areas at Different Latitudes

We need to compare the areas of three strips:

• $A_1$: At the Equator ($\phi = 0^\circ$). Here, $\cos(0^\circ) = 1$.
• $A_2$: At the Tropic of Cancer ($\phi \approx 23.5^\circ$). Here, $\cos(23.5^\circ) \approx 0.917$.
• $A_3$: At the Arctic Circle ($\phi \approx 66.5^\circ$). Here, $\cos(66.5^\circ) \approx 0.397$.

The cosine function decreases as the angle increases from $0^\circ$ to $90^\circ$. Therefore:

$ \cos(0^\circ) > \cos(23.5^\circ) > \cos(66.5^\circ) $

This implies:

$ 1 > 0.917 > 0.397 $

Since area is proportional to $\cos(\phi)$, the order of the areas is:

$ A_1 > A_2 > A_3 $

Conclusion

The strip at the Equator has the largest area ($A_1$), followed by the strip at the Tropic of Cancer ($A_2$), and the strip at the Arctic Circle has the smallest area ($A_3$).