A 7.5 minute quadrangle covers land at a scale of 1:24000 and spans 7.5 minutes of arc in each direction. Approximately how many square miles does it cover?

A 7.5 minute quadrangle covers land at a scale of 1:24000 and spans 7.5 minutes of arc in each direction. Approximately how many square miles does it cover?

• A. 49-70 square miles
• B. 10-20 square miles
• C. 100-120 square miles
• D. 200-250 square miles

Answer: (A)

Think in terms of how far a 7.5-minute span of arc is on the ground. A 7.5-minute quadrangle spans 7.5 minutes of latitude and 7.5 minutes of longitude.

A minute of latitude is about 1.15 miles (since 1 degree of latitude is roughly 69 miles). So the vertical extent is 7.5 × 1.15 ≈ 8.6 miles.

A minute of longitude is smaller by the cosine of the latitude. At mid-latitudes, cos(latitude) is about 0.75–0.77, so a minute of longitude is roughly 1.15 × cos(latitude) miles. The horizontal extent is 7.5 × 1.15 × cos(latitude) ≈ 8.6 × cos(latitude), which comes out around 6.1–6.6 miles.

Multiplying the two sides gives an area near 8.6 × 6.4 ≈ 55 square miles (roughly 50–60). A value around 57 square miles falls in the 49–70 square mile range, so that’s the best match.