Scale at any latitude (λ) = Scale at Equator × sec (λ) — Page 313, Lesson 279

I want to walk you through the core formula that governs scale on a Mercator chart. This is the starting point for every Mercator scale problem you will solve. The formula is: Scale at any latitude (λ) = Scale at the Equator × sec (λ) Here, λ is the Greek letter lambda, which stands for latitude. "sec" is short for secant, which is a trigonometric function. If you prefer, you can always rewrite secant as 1 divided by cosine of that angle. So the same formula can be written as: Scale at any latitude = Scale at the Equator × (1 / cos λ). This relationship is the foundation. Questions on this topic generally fall into four basic types. First, given the scale at the Equator, find the Mercator scale at some other latitude. Second, given the Mercator scale at some latitude, find the scale at the Equator. Third, given the Mercator scale at one latitude, find the Mercator scale at another latitude. And fourth, given a fixed chart distance between meridians, find the Mercator scale at a specific latitude. Let's work through the first type with an example. Example 1: If the scale of a Mercator chart at the Equator is 1:1,000,000, what is the scale at 60° North or South? The answer choices are: a) 1:2,000,000, b) 1:1,000,000, c) 1:866,000, or d) 1:500,000. Whenever you solve problems using representative fractions — which we write as RF — remember that the scale must be written mathematically in fraction form. So we set up the equation: Scale at 60°N = Scale at Equator × secant 60° That gives us: Scale at 60°N = (1 / 1,000,000) × sec 60° Now, it's easier if we write secant as 1/cos. That way we keep 1 as the numerator, which is what we want for a representative fraction, and we only have to multiply out the denominators. So: Scale at 60°N = (1 / 1,000,000) × (1 / cos 60°) cos 60° equals 0.5. So: Scale at 60°N = (1 / 1,000,000) × (1 / 0.5) 1 divided by 0.5 is 2. So: Scale at 60°N = (1 / 1,000,000) × 2 = 2 / 1,000,000 That simplifies to 1 / 500,000. So the answer is 1:500,000, which is option d. Here's the important trap to watch for: at 60° North or South, the scale on a Mercator chart is double the scale at the Equator. If the scale doubles, the denominator of the representative fraction is halved. Remember this rule: if scale expands, the denominator decreases; if scale contracts, the denominator increases. Now let's look at the second type of problem: given Mercator scale at some latitude, find the scale at the Equator. Example 2: If the scale of a Mercator chart at 52° South is 1:2,000,000, what is the scale at the Equator? The answer choices are: a) 1:3,250,000, b) 1:1,000,000, c) 1:866,000, or d) 1:500,000. We start the same way: Scale at 52°S = Scale at Equator × secant 52°S Now substitute what we know: 1 / 2,000,000 = Scale at Equator × (1 / cos 52°) We need to rearrange to solve for Scale at Equator. Multiply both sides by cos 52°: Scale at Equator = (cos 52°) / 2,000,000 But wait — that gives us a fraction with a numerator that isn't 1. We want a representative fraction with 1 as the numerator. So it's easier to enter this into your calculator denominator first, then divide by the numerator. That gives you the value of the denominator with 1 as the numerator. Key in: 2,000,000 ÷ cos 52 = cos 52° is approximately 0.61566. So 2,000,000 divided by 0.61566 gives you 3,248,538. You write that as: 1 / 3,248,538 That rounds to 1:3,250,000, which is option a. So the key takeaway: the Mercator scale formula is your starting point. Write secant as 1/cos to keep the numerator as 1. And remember that scale increases as you move away from the Equator, so the denominator gets smaller.